share/contrib/diffequations/tests/rtestode_kamke_2_2.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Second order linear equations 101-200
   5
   6   Reference:
   7     E Kamke, Differentialgleichungen Losungsmethoden und Losungen,
   8     Vol 1, Geest & Portig, Leipzig, 1961
   9 */
  10
  11 /* Print ode number*/
  12 (pn_(n_):=print("Kamke ODE 2.",n_),true);
  13 true;
  14
  15 /* Kamke 2. 101  */
  16 (pn_(101),ans:contrib_ode(eqn:x*'diff(y,x,2)+2*'diff(y,x,1)+a*x*y,y,x));
  17 [y = %k2*%e^(sqrt(-a)*x)/x+%k1*%e^-(sqrt(-a)*x)/x];
  18 [method, ode_check(eqn,ans[1])];
  19 [odelin,0];
  20
  21 /* Kamke 2. 102  */
  22 (pn_(102),ans:contrib_ode(eqn:x*'diff(y,x,2)+2*'diff(y,x,1)+a*x^2*y,y,x),0);
  23 0$
  24 /*
  25 [y = bessel_y(1/3,2*sqrt(a)*x^(3/2)/3)*%k2/sqrt(x)
  26     +bessel_j(1/3,2*sqrt(a)*x^(3/2)/3)*%k1/sqrt(x)];
  27  */
  28 [method,ode_check(eqn,ans[1])];
  29 [odelin,0];
  30
  31 /* Kamke 2. 103  */
  32 (pn_(103),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*'diff(y,x,1)+a*y,y,x));
  33 [y = -bessel_y(3,2*sqrt(a)*sqrt(x))*%k2*x^(3/2)
  34      -bessel_j(3,2*sqrt(a)*sqrt(x))*%k1*x^(3/2)];
  35 [method,ode_check(eqn,ans[1])];
  36 [odelin,0];
  37
  38 /* Kamke 2. 104  */
  39 (pn_(104),ans:contrib_ode(eqn:x*'diff(y,x,2)+v*'diff(y,x,1)+a*y,y,x));
  40 [y = %k2*bessel_y(v-1,2*sqrt(a)*sqrt(x))/x^(v/2-1/2)
  41     +%k1*bessel_j(v-1,2*sqrt(a)*sqrt(x))/x^(v/2-1/2)];
  42 [method,ode_check(eqn,ans[1])];
  43 [odelin,0];
  44
  45 /* Kamke 2. 105  */
  46 (pn_(105),ans:contrib_ode(eqn:x*'diff(y,x,2)+a*'diff(y,x,1)+b*x*y,y,x),0);
  47 0$
  48 /*
  49  [y = %k2*bessel_y(1/2-(a/2),-sqrt(b)*x)*x^(1/2-(a/2))
  50      +%k1*bessel_j(1/2-(a/2),-sqrt(b)*x)*x^(1/2-(a/2))];
  51  */
  52 [method,ode_check(eqn,ans[1])];
  53 [odelin,0];
  54
  55 /* Kamke 2. 106  */
  56 (pn_(106),ans:contrib_ode(eqn:x*'diff(y,x,2)+a*'diff(y,x,1)+b*x^a1*y,y,x));
  57 false;
  58
  59 /* Kamke 2. 107  */
  60 (pn_(107),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x+b)*'diff(y,x,1)+a*y,y,x));
  61 [y = %k2*kummer_u(a,b,-x)+%k1*kummer_m(a,b,-x)];
  62 [method, ode_check(eqn,ans[1])];
  63 [odelin,0];
  64
  65 /* Kamke 2. 108  */
  66 (pn_(108),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x+b+a)*'diff(y,x,1)+a*y,y,x));
  67 [y = %k2*kummer_u(a,b+a,-x)+%k1*kummer_m(a,b+a,-x)];
  68 [method, ode_check(eqn,ans[1])];
  69 [odelin,0];
  70
  71 /* Kamke 2. 109  */
  72 (pn_(109),ans:contrib_ode(eqn:x*'diff(y,x,2)-x*'diff(y,x,1)-y-x*(x+1)*%e^x,y,x));
  73 [y = x*%e^x*'integrate((gamma_incomplete(-1,x)*x^2+gamma_incomplete(-1,x)*x)*%e^x,x)
  74     +(-gamma_incomplete(-1,x)*x^3+gamma_incomplete(-1,x)*x^2-gamma_incomplete(-1,x)*x)
  75     *%e^(2*x)+%k2*x*%e^x-gamma_incomplete(-1,x)*%k1*x*%e^x];
  76 ode_check(eqn,ans[1]);
  77 0;
  78 method;
  79 variationofparameters;
  80
  81 /* Kamke 2. 110 */
  82 assume(x>0,x+4*a>0);
  83 [x>0,x+4*a>0];
  84 (pn_(110),ans:contrib_ode(eqn:x*'diff(y,x,2)-x*'diff(y,x,1)-a*y,y,x));
  85 [y = %k2*kummer_u(-a,0,-x)*%e^x+%k1*kummer_m(-a,0,-x)*%e^x];
  86 [method,ode_check(eqn,ans[1])];
  87 [odelin,0];
  88 forget(x>0,x+4*a>0);
  89 [x>0,x+4*a>0];
  90
  91 /* Kamke 2. 111 */
  92 (pn_(111),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x+1)*'diff(y,x,1)+y,y,x));
  93 [y = %k2*%e^x+%k1*(x+1)];
  94 [method,ode_check(eqn,ans[1])];
  95 [odelin,0];
  96
  97 /* Kamke 2. 112  */
  98 (pn_(112),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x+1)*'diff(y,x,1)-2*(x-1)*y,y,x));
  99 [y = %k2*%e^(2*x)+%k1*(3*x+1)*%e^-x]$
 100 [method, ode_check(eqn,ans[1])];
 101 [odelin,0];
 102
 103 /* Kamke 2. 113  */
 104 assume(x^2+(4*a-2*b)*x+b^2>0);
 105 [x^2+(4*a-2*b)*x+b^2>0];
 106 (pn_(113),ans:contrib_ode(eqn:x*'diff(y,x,2)+(b-x)*'diff(y,x,1)-a*y,y,x));
 107 [y = %k2*kummer_u(b-a,b,-x)*%e^x+%k1*kummer_m(b-a,b,-x)*%e^x];
 108 [method,ode_check(eqn,ans[1])];
 109 [odelin,0];
 110 forget(x^2+(4*a-2*b)*x+b^2>0);
 111 [x^2+(4*a-2*b)*x+b^2>0];
 112
 113 /* Kamke 2. 114  */
 114 (pn_(114),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*(x-1)*'diff(y,x,1)-y,y,x));
 115 [y = kummer_u(1/2,0,-2*x)*%k2*%e^(2*x)/x
 116     +kummer_m(1/2,0,-2*x)*%k1*%e^(2*x)/x];
 117 [method, ode_check(eqn,ans[1])];
 118 [odelin,0];
 119
 120 /* Kamke 2. 115  */
 121 (pn_(115),ans:contrib_ode(eqn:x*'diff(y,x,2)-(3*x-2)*'diff(y,x,1)+(3-2*x)*y,y,x));
 122 [y = kummer_u(6*17^-(1/2),0,-sqrt(17)*x)*%k2*%e^(sqrt(17)*x/2+3*x/2)/x
 123     +kummer_m(6*17^-(1/2),0,-sqrt(17)*x)*%k1*%e^(sqrt(17)*x/2+3*x/2)/x];
 124 [method, ode_check(eqn,ans[1])];
 125 [odelin,0];
 126
 127 /* Kamke 2. 116  */
 128 (pn_(116),ans:contrib_ode(eqn:x*'diff(y,x,2)+(a*x+n+b)*'diff(y,x,1)+a*n*y,y,x));[y = %k2*kummer_u(n,n+b,-a*x)+%k1*kummer_m(n,n+b,-a*x)];
 129 [method, ode_check(eqn,ans[1])];
 130 [odelin,0];
 131
 132 /* Kamke 2. 117  */
 133 (pn_(117),ans:contrib_ode(eqn:x*'diff(y,x,2)-(b+a)*(x+1)*'diff(y,x,1)+a*b*x*y,y,x),0);
 134 0$
 135 /*
 136 [y = %k2*kummer_u(-(b^2/(b-a))-(a*b/(b-a)),-b-a,a*x-b*x)*%e^(b*x)
 137     +%k1*kummer_m(-(b^2/(b-a))-(a*b/(b-a)),-b-a,a*x-b*x)*%e^(b*x)];
 138  */
 139 [method, ode_check(eqn,ans[1])];
 140 [odelin,0];
 141
 142 /* Kamke 2. 118  */
 143 (pn_(118),ans:contrib_ode(eqn:x*'diff(y,x,2)+((b+a)*x+n+m)*'diff(y,x,1)+(a*b*x+a*n+b*m)*y,y,x),0);
 144 0$
 145 /*
 146 [y = %k2*kummer_u(m,n+m,a*x-b*x)*%e^-(a*x)
 147     +%k1*kummer_m(m,n+m,a*x-b*x)*%e^-(a*x)];
 148  */
 149 [method, ode_check(eqn,ans[1])];
 150 [odelin,0];
 151
 152 /* Kamke 2. 119  */
 153 /* This causes 2.160 to fail for no obvious reason.  Result is OK */
 154 /*
 155 assume(notequal(b,-1/2));
 156 [notequal(b,-1/2)];
 157 (pn_(119),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*(a*x+b)*'diff(y,x,1)+(a^2*x+2*a*b)*y,y,x));
 158 [y = %k2*x^(2*b+1)*%e^(a*x)+%k1*%e^(a*x)];
 159 [method,ode_check(eqn,ans[1])];
 160 [odelin,0];
 161 forget(notequal(b,-1/2));
 162 [notequal(b,-1/2)];
 163 */
 164
 165 /* Kamke 2. 120  */
 166 (pn_(120),ans:contrib_ode(eqn:x*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+(c*x+d)*y,y,x),0);
 167 0$
 168 /*
 169 [y = %k2*kummer_u(-(2*sqrt(a^2-4*c)*d/(8*c-2*a^2))
 170                          +4*b*c/(8*c-2*a^2)+a*b*sqrt(a^2-4*c)/(8*c-2*a^2)
 171                          -(a^2*b/(8*c-2*a^2)),b,-sqrt(a^2-4*c)*x)
 172               *%e^(sqrt(a^2-4*c)*x/2-(a*x/2))
 173     +%k1*kummer_m(-(2*sqrt(a^2-4*c)*d/(8*c-2*a^2))
 174                         +4*b*c/(8*c-2*a^2)+a*b*sqrt(a^2-4*c)/(8*c-2*a^2)
 175                         -(a^2*b/(8*c-2*a^2)),b,-sqrt(a^2-4*c)*x)
 176              *%e^(sqrt(a^2-4*c)*x/2-(a*x/2))] ;
 177  */
 178 [method, ode_check(eqn,ans[1])];
 179 [odelin,0];
 180
 181 /* Kamke 2. 121  */
 182 (pn_(121),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x^2-x)*'diff(y,x,1)+(x-1)*y,y,x));
 183 [y = %k2*x*'integrate(%e^(x^2/2-x)/x^2,x)+%k1*x];
 184 [method, ode_check(eqn,ans[1])];
 185 [odelin,0];
 186
 187 /* Kamke 2. 122  */
 188 (pn_(122),ans:contrib_ode(eqn:x*'diff(y,x,2)-(x^2-x-2)*'diff(y,x,1)-x*(x+3)*y,y,x));
 189 [y = %k2*%e^(x^2/2)*'integrate(%e^(-x^2/2-x)/x^2,x)+%k1*%e^(x^2/2)];
 190 [method, ode_check(eqn,ans[1])];
 191 [odelin,0];
 192
 193 /* Kamke 2. 123  */
 194 assume(b>0,b>a^2);
 195 [b>0,b>a^2];
 196 (pn_(123),ans:contrib_ode(eqn:x*'diff(y,x,2)-(2*a*x^2+1)*'diff(y,x,1)+b*x^3*y,y,x));
 197 [y=%e^(a*x^2/2)*(%k1*sin(sqrt(4-4*a^2/b)*sqrt(b)*x^2/4)+%k2*cos(sqrt(4-4*a^2/b)*sqrt(b)*x^2/4))];
 198 ode_check(eqn,ans[1]);
 199 0;
 200 method;
 201 xformtoconstcoeff;
 202 forget(b>0,b>a^2);
 203 [b>0,b>a^2];
 204
 205 /* Kamke 2. 124  */
 206 (pn_(124),ans:contrib_ode(eqn:x*'diff(y,x,2)-2*(x^2-a)*'diff(y,x,1)+2*n*x*y,y,x),0);
 207 0$
 208 /*
 209 [y = %k2*kummer_u(-(n/2),a+1/2,x^2)+%k1*kummer_m(-(n/2),a+1/2,x^2)];
 210  */
 211 [method, ode_check(eqn,ans[1])];
 212 [odelin,0];
 213
 214 /* Kamke 2. 125  */
 215 (pn_(125),ans:contrib_ode(eqn:x*'diff(y,x,2)+(4*x^2-1)*'diff(y,x,1)-4*x^3*y-4*x^5,y,x));
 216 [y=%e^-x^2*(%i*%k1*sinh(sqrt(2)*x^2)+%k2*cosh(sqrt(2)*x^2))-%e^-(sqrt(2)*x^2)*((((sqrt(2)-2)*x^2+3*sqrt(2)-4)*%e^(2*sqrt(2)*x^2)+(sqrt(2)+2)*x^2+3*sqrt(2)+4)*sinh(sqrt(2)*x^2)+(((2-sqrt(2))*x^2-3*sqrt(2)+4)*%e^(2*sqrt(2)*x^2)+(sqrt(2)+2)*x^2+3*sqrt(2)+4)*cosh(sqrt(2)*x^2))/4];
 217 [method,ode_check(eqn,ans[1])];
 218 [variationofparameters,0];
 219
 220 /* Kamke 2. 126  */
 221 /* CHECKME: Takes a long time
 222 (pn_(126),ans:contrib_ode(eqn:x*'diff(y,x,2)+(2*a*x^3-1)*'diff(y,x,1)+(a^2*x^3+a)*y,y,x));
 223 false;
 224 */
 225
 226 /* Kamke 2. 127  */
 227 (pn_(127),ans:contrib_ode(eqn:x*'diff(y,x,2)+(2*a*x*log(x)+1)*'diff(y,x,1)+(a^2*x*log(x)^2+a*log(x)+a)*y,y,x));
 228 false;
 229
 230 /* Kamke 2. 128  */
 231 (pn_(128),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x*f(x)+2)*'diff(y,x,1)+f(x)*y,y,x));
 232 false;
 233
 234 /* Kamke 2. 129  */
 235 (pn_(129),ans:contrib_ode(eqn:(x-3)*'diff(y,x,2)-(4*x-9)*'diff(y,x,1)+(3*x-6)*y,y,x));
 236 [y = %k2*(4*x^3-42*x^2+150*x-183)*%e^(3*x)+%k1*%e^x]$
 237 [method, ode_check(eqn,ans[1])];
 238 [odelin,0];
 239
 240 /* Kamke 2. 130  */
 241 (pn_(130),ans:contrib_ode(eqn:2*x*'diff(y,x,2)+'diff(y,x,1)+a*y,y,x));
 242 [y=%k1*sin(2*sqrt(a)*sqrt(x)/sqrt(2))+%k2*cos(2*sqrt(a)*sqrt(x)/sqrt(2))];
 243 ode_check(eqn,ans[1]);
 244 0;
 245 method;
 246 xformtoconstcoeff;
 247
 248 /* Kamke 2. 131  */
 249 (pn_(131),ans:contrib_ode(eqn:2*x*'diff(y,x,2)-(x-1)*'diff(y,x,1)+a*y,y,x));
 250 [y = %k2*kummer_u(a+1,3/2,-x/2)*sqrt(x)*%e^(x/2)
 251     +%k1*kummer_m(a+1,3/2,-x/2)*sqrt(x)*%e^(x/2)];
 252 [method, ode_check(eqn,ans[1])];
 253 [odelin,0];
 254
 255 /* Kamke 2. 132  */
 256 assume(4*x^2+(-8*a-4)*x+1>0);
 257 [4*x^2+(-8*a-4)*x+1>0];
 258 (pn_(132),ans:contrib_ode(eqn:2*x*'diff(y,x,2)-(2*x-1)*'diff(y,x,1)+a*y,y,x),0);
 259 0$
 260 /*
 261 [y = %k2*kummer_u(a/2+1,3/2,-x)*sqrt(x)*%e^x
 262     +%k1*kummer_m(a/2+1,3/2,-x)*sqrt(x)*%e^x];
 263  */
 264 [method, ode_check(eqn,ans[1])];
 265 [odelin,0];
 266 forget(4*x^2+(-8*a-4)*x+1>0);
 267 [4*x^2+(-8*a-4)*x+1>0];
 268
 269 /* Kamke 2. 133  */
 270 (pn_(133),ans:contrib_ode(eqn:(2*x-1)*'diff(y,x,2)-(3*x-4)*'diff(y,x,1)+(x-3)*y,y,x));
 271 [y = %k2*%e^x+gamma_incomplete(-1/4,(2*x-1)/4)*%k1*%e^(x-1/4)];
 272 [method, ode_check(eqn,ans[1])];
 273 [odelin,0];
 274
 275 /* Kamke 2. 134  */
 276 (pn_(134),ans:contrib_ode(eqn:4*x*'diff(y,x,2)-(x+a)*y,y,x));
 277 [y = %k2*kummer_u(-a/4,0,-x)*%e^(x/2)+%k1*kummer_m(-a/4,0,-x)*%e^(x/2)];
 278 [method, ode_check(eqn,ans[1])];
 279 [odelin,0];
 280
 281 /* Kamke 2. 135  */
 282 (pn_(135),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+2*'diff(y,x,1)-y,y,x));
 283 [y=%i*%k1*sinh(sqrt(x))+%k2*cosh(sqrt(x))];
 284 ode_check(eqn,ans[1]);
 285 0;
 286 method;
 287 xformtoconstcoeff;
 288
 289 /* Kamke 2. 136  */
 290 (pn_(136),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+4*'diff(y,x,1)+(-x-2)*y,y,x));
 291 [y = gamma_incomplete(0,x)*%k2*%e^(x/2)+%k1*%e^(x/2)];
 292 [method, ode_check(eqn,ans[1])];
 293 [odelin,0];
 294
 295 /* Kamke 2. 137  */
 296 (pn_(137),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+(-x-2)*y+L*y+4*y,y,x),0);
 297 0$
 298 /*
 299 [y = %k2*kummer_u(L/4+1/2,0,-x)*%e^(x/2)
 300     +%k1*kummer_m(L/4+1/2,0,-x)*%e^(x/2)];
 301  */
 302 [method, ode_check(eqn,ans[1])];
 303 [odelin,0];
 304
 305 /* Kamke 2. 138  */
 306 (pn_(138),ans:contrib_ode(eqn:4*x*'diff(y,x,2)+4*m*'diff(y,x,1)+(-x+4*n+2*m)*y,y,x));
 307 [y = %k2*kummer_u(n+m,m,-x)*%e^(x/2)+%k1*kummer_m(n+m,m,-x)*%e^(x/2)];
 308 [method, ode_check(eqn,ans[1])];
 309 [odelin,0];
 310
 311 /* Kamke 2. 139  */
 312 (pn_(139),ans:contrib_ode(eqn:16*x*'diff(y,x,2)+8*'diff(y,x,1)+(-x-a)*y,y,x),0);
 313 0$
 314 /*
 315 [y = %k2*kummer_u(3/4-(a/8),3/2,-(x/2))*sqrt(x)*%e^(x/4)
 316     +%k1*kummer_m(3/4-(a/8),3/2,-(x/2))*sqrt(x)*%e^(x/4)];
 317  */
 318 [method, ode_check(eqn,ans[1])];
 319 [odelin,0];
 320
 321 /* Kamke 2. 140  */
 322 (pn_(140),ans:contrib_ode(eqn:a*x*'diff(y,x,2)+b*'diff(y,x,1)+c*y,y,x),0);
 323 0$
 324 /*
 325 [y = %k2*bessel_y(b/a-1,2*sqrt(c)*sqrt(x)/sqrt(a))*x^(1/2-(b/(2*a)))
 326     +%k1*bessel_j(b/a-1,2*sqrt(c)*sqrt(x)/sqrt(a))*x^(1/2-(b/(2*a)))];
 327  */
 328 [method, ode_check(eqn,ans[1])];
 329 [odelin,0];
 330
 331 /* Kamke 2. 141  */
 332 (pn_(141),ans:contrib_ode(eqn:a*x*'diff(y,x,2)+(b*x+3*a)*'diff(y,x,1)+3*b*y,y,x));
 333 [y = gamma_incomplete(-2,-b*x/a)*%k2*%e^-(b*x/a)+%k1*%e^-(b*x/a)];
 334 [method, ode_check(eqn,ans[1])];
 335 [odelin,0];
 336
 337 /* Kamke 2. 142  */
 338 (pn_(142),ans:contrib_ode(eqn:5*(a*x+b)*'diff(y,x,2)+8*a*'diff(y,x,1)+c*(a*x+b)^(1/5)*y,y,x));
 339 false;
 340
 341 /* Kamke 2. 143  */
 342 (pn_(143),ans:contrib_ode(eqn:2*a*x*'diff(y,x,2)+(b*x+a)*'diff(y,x,1)+c*y,y,x));
 343 [y = %k2*kummer_u(c/b,1/2,-b*x/(2*a))+%k1*kummer_m(c/b,1/2,-b*x/(2*a))];
 344 [method, ode_check(eqn,ans[1])];
 345 [odelin,0];
 346
 347 /* Kamke 2. 144  */
 348 (pn_(144),ans:contrib_ode(eqn:2*a*x*'diff(y,x,2)+(b*x+3*a)*'diff(y,x,1)+c*y,y,x));
 349 [y = %k2*kummer_u(c/b,3/2,-b*x/(2*a))+%k1*kummer_m(c/b,3/2,-b*x/(2*a))];
 350 [method, ode_check(eqn,ans[1])];
 351 [odelin,0];
 352
 353 /* Kamke 2. 145  */
 354 (pn_(145),ans:contrib_ode(eqn:(a2*x+b2)*'diff(y,x,2)+(a1*x+b1)*'diff(y,x,1)+(a0*x+b0)*y,y,x),0);
 355 0$
 356 /*
 357 [y = %k2*kummer_u(2*a0*a2*sqrt(a1^2-4*a0*a2)*b2/(8*a0*a2^3-2*a1^2*a2^2)
 358                           -(a1^2*sqrt(a1^2-4*a0*a2)*b2/(8*a0*a2^3-2*a1^2*a2^2))
 359                           -(4*a0*a1*a2*b2/(8*a0*a2^3-2*a1^2*a2^2))
 360                           +a1^3*b2/(8*a0*a2^3-2*a1^2*a2^2)
 361                           +a1*a2*sqrt(a1^2-4*a0*a2)*b1/(8*a0*a2^3-2*a1^2*a2^2)
 362                           +4*a0*a2^2*b1/(8*a0*a2^3-2*a1^2*a2^2)
 363                           -(a1^2*a2*b1/(8*a0*a2^3-2*a1^2*a2^2))
 364                           -(2*a2^2*sqrt(a1^2-4*a0*a2)*b0
 365                            /(8*a0*a2^3-2*a1^2*a2^2)),b1/a2-(a1*b2/a2^2),
 366                          -(sqrt(a1^2-4*a0*a2)*x/a2)-(sqrt(a1^2-4*a0*a2)*b2/a2^2))
 367                *%e^(sqrt(a1^2-4*a0*a2)*x/(2*a2)-(a1*x/(2*a2))
 368                                                +sqrt(a1^2-4*a0*a2)*b2
 369                                                 /(2*a2^2))
 370           +%k1*kummer_m(2*a0*a2*sqrt(a1^2-4*a0*a2)*b2/(8*a0*a2^3-2*a1^2*a2^2)
 371                          -(a1^2*sqrt(a1^2-4*a0*a2)*b2/(8*a0*a2^3-2*a1^2*a2^2))
 372                          -(4*a0*a1*a2*b2/(8*a0*a2^3-2*a1^2*a2^2))
 373                          +a1^3*b2/(8*a0*a2^3-2*a1^2*a2^2)
 374                          +a1*a2*sqrt(a1^2-4*a0*a2)*b1/(8*a0*a2^3-2*a1^2*a2^2)
 375                          +4*a0*a2^2*b1/(8*a0*a2^3-2*a1^2*a2^2)
 376                          -(a1^2*a2*b1/(8*a0*a2^3-2*a1^2*a2^2))
 377                          -(2*a2^2*sqrt(a1^2-4*a0*a2)*b0
 378                           /(8*a0*a2^3-2*a1^2*a2^2)),b1/a2-(a1*b2/a2^2),
 379                         -(sqrt(a1^2-4*a0*a2)*x/a2)-(sqrt(a1^2-4*a0*a2)*b2/a2^2))
 380               *%e^(sqrt(a1^2-4*a0*a2)*x/(2*a2)-(a1*x/(2*a2))
 381                                               +sqrt(a1^2-4*a0*a2)*b2
 382                                                /(2*a2^2))];
 383  */
 384 [method, ode_check(eqn,ans[1])];
 385 [odelin,0];
 386
 387 /* Kamke 2. 146  */
 388 assume(x>0);
 389 [x>0];
 390 (pn_(146),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-6*y,y,x));
 391 [y=%e^(log(x)/2)*(%i*%k1*sinh(5*log(x)/2)+%k2*cosh(5*log(x)/2))];
 392 ode_check(eqn,ans[1]);
 393 0;
 394 method;
 395 xformtoconstcoeff;
 396 forget(x>0);
 397 [x>0];
 398
 399 /* Kamke 2. 147  */
 400 assume(x>0);
 401 [x>0];
 402 (pn_(147),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-12*y,y,x));
 403 [y=%e^(log(x)/2)*(%i*%k1*sinh(7*log(x)/2)+%k2*cosh(7*log(x)/2))];
 404 ode_check(eqn,ans[1]);
 405 0;
 406 method;
 407 xformtoconstcoeff;
 408 forget(x>0);
 409 [x>0];
 410
 411 /* Kamke 2. 148  */
 412 assume(a>0,a>1/4);
 413 [a>0,a>1/4];
 414 (pn_(148),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*y,y,x));
 415 [y=%e^(log(x)/2)*(%k1*sin(sqrt(4-1/a)*sqrt(a)*log(x)/2)+%k2*cos(sqrt(4-1/a)*sqrt(a)*log(x)/2))];
 416 ode_check(eqn,ans[1]);
 417 0;
 418 method;
 419 xformtoconstcoeff;
 420 forget(a>0,a>1/4);
 421 [a>0,a>1/4];
 422
 423 /* Kamke 2. 149  */
 424 (pn_(149),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x+b)*y,y,x));
 425 [y = %k2*bessel_y(sqrt(1-4*b),2*sqrt(a)*sqrt(x))*sqrt(x)
 426     +%k1*bessel_j(sqrt(1-4*b),2*sqrt(a)*sqrt(x))*sqrt(x)];
 427 [method,ode_check(eqn,ans[1])];
 428 [odelin,0];
 429
 430 /* Kamke 2. 150  */
 431 (pn_(150),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x^2-2)*y,y,x));
 432 [y = bessel_y(-3/2,-x)*%k2*sqrt(-x)+bessel_j(-3/2,-x)*%k1*sqrt(-x)];
 433 [method,ode_check(eqn,ans[1])];
 434 [odelin,0];
 435
 436 /* Kamke 2. 151  */
 437 (pn_(151),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(a*x^2+2)*y,y,x));
 438 [y = bessel_y(-3/2,-sqrt(-a)*x)*%k2*sqrt(-x)
 439     +bessel_j(-3/2,-sqrt(-a)*x)*%k1*sqrt(-x)];
 440 [method, ode_check(eqn,ans[1])];
 441 [odelin,0];
 442
 443 /* Kamke 2. 152  */
 444 (pn_(152),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a^2*x^2-6)*y,y,x));
 445 [y = bessel_y(5/2,-a*x)*%k2*sqrt(-x)+bessel_j(5/2,-a*x)*%k1*sqrt(-x)];
 446 [method, ode_check(eqn,ans[1])];
 447 [odelin,0];
 448
 449 /* Kamke 2. 153  */
 450 (pn_(153),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^2-(v-1)*v)*y,y,x),0);
 451 0$
 452 /*
 453 [y = %k2*bessel_y(1/2-v,-sqrt(a)*x)*sqrt(-x)
 454     +%k1*bessel_j(1/2-v,-sqrt(a)*x)*sqrt(-x)];
 455  */
 456 [method, ode_check(eqn,ans[1])];
 457 [odelin,0];
 458
 459 /* Kamke 2. 154  */
 460 (pn_(154),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^2+b*x+c)*y,y,x),0);
 461 0$
 462 /*
 463 [y = %k2*kummer_u(sqrt(1-4*c)/2-(sqrt(-a)*b/(2*a))+1/2,sqrt(1-4*c)+1,
 464                          -2*sqrt(-a)*x)*(-x)^(sqrt(1-4*c)/2+1/2)
 465               *%e^(sqrt(-a)*x)
 466     +%k1*kummer_m(sqrt(1-4*c)/2-(sqrt(-a)*b/(2*a))+1/2,sqrt(1-4*c)+1,
 467                         -2*sqrt(-a)*x)*(-x)^(sqrt(1-4*c)/2+1/2)
 468               *%e^(sqrt(-a)*x)];
 469  */
 470 [method, ode_check(eqn,ans[1])];
 471 [odelin,0];
 472
 473 /* Kamke 2. 155  */
 474 (pn_(155),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x^k-(b-1)*b)*y,y,x));
 475 false;
 476
 477 /* Kamke 2. 156  */
 478 (pn_(156),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+y/log(x)-x*%e^x*(x*log(x)+2),y,x));
 479 false;
 480
 481 /* Kamke 2. 157  */
 482 (pn_(157),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*'diff(y,x,1)-x*y,y,x));
 483 false;
 484
 485 /* Kamke 2. 158  */
 486 (pn_(158),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*'diff(y,x,1)+(-b^2*x^2-a*b)*y,y,x));
 487 [y = %k2*%e^(b*x)*'integrate(%e^(a/x-2*b*x),x)+%k1*%e^(b*x)];
 488 [method, ode_check(eqn,ans[1])];
 489 [odelin,0];
 490
 491 /* Kamke 2. 159  */
 492 (pn_(159),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)-y-a*x^2,y,x));
 493 [y=a*x^2/3+%k2*x-%k1/(2*x)];
 494 ode_check(eqn,ans[1]);
 495 0;
 496 method;
 497 variationofparameters;
 498
 499 /* Kamke 2. 160  */
 500 assume(a>0);
 501 [a>0];
 502 (pn_(160),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)+a*y,y,x));
 503 [y=%k1*sin(sqrt(4*a)*log(x)/2)+%k2*cos(sqrt(4*a)*log(x)/2)];
 504 ode_check(eqn,ans[1]);
 505 0;
 506 method;
 507 euler;
 508 forget(a>0);
 509 [a>0];
 510
 511 /* Kamke 2. 161  */
 512 (pn_(161),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)+(-x-a)*y,y,x));
 513 [y=%k2*bessel_y(2*sqrt(a),2*sqrt(-x))+%k1*bessel_j(2*sqrt(a),2*sqrt(-x))];
 514 [method, ode_check(eqn,ans[1])];
 515 [odelin,0];
 516
 517 /* Kamke 2. 162  */
 518 declare(v,integer);
 519 done;
 520 (pn_(162),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)+(x^2-v^2)*y,y,x));
 521 [y = %k2*bessel_y(v,x)+%k1*bessel_j(v,x)];
 522 [method,ode_check(eqn,ans[1])];
 523 [bessel,0];
 524 remove(v,integer);
 525 done;
 526
 527 /* Kamke 2. 163  Note: this is still correct for f(x)=0 */
 528 declare(v,integer);
 529 done;
 530 (pn_(163),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)+(x^2-v^2)*y-f(x),y,x));
 531 [y = 2*bessel_j(v,x)*'integrate(bessel_y(v,x)*f(x)
 532    /((bessel_j(v,x)*bessel_y(v+1,x)-bessel_y(v,x)*bessel_j(v+1,x)
 533      +bessel_j(v-1,x)*bessel_y(v,x)-bessel_y(v-1,x)*bessel_j(v,x))*x^2),x)
 534     -2*bessel_y(v,x)*'integrate(bessel_j(v,x)*f(x)
 535    /((bessel_j(v,x)*bessel_y(v+1,x)-bessel_y(v,x)*bessel_j(v+1,x)
 536      +bessel_j(v-1,x)*bessel_y(v,x)-bessel_y(v-1,x)*bessel_j(v,x))*x^2),x)
 537  +%k2*bessel_y(v,x)+%k1*bessel_j(v,x)];
 538 [method,ode_check(eqn,ans[1])];
 539 [variationofparameters,0];
 540 remove(v,integer);
 541 done;
 542
 543 /* Kamke 2. 164  */
 544 (pn_(164),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*'diff(y,x,1)+(l*x^2-v^2)*y,y,x));
 545 [y = %k2*bessel_y(v,-sqrt(l)*x)+%k1*bessel_j(v,-sqrt(l)*x)];
 546 [method, ode_check(eqn,ans[1])];
 547 [odelin,0];
 548
 549 /* Kamke 2. 165  */
 550 (pn_(165),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x+a)*'diff(y,x,1)-y,y,x));
 551 [y = gamma_incomplete(2,a/x)*%k1*x*%e^(a/x)/a^2+%k2*x*%e^(a/x)];
 552 ode_check(eqn,ans[1]);
 553 0;
 554 method;
 555 exact;
 556
 557 /* Kamke 2. 166  */
 558 (pn_(166),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-x*'diff(y,x,1)+y-3*x^3,y,x));
 559 [y=x*(%k2*log(x)+%k1)+3*x^3/4];
 560 ode_check(eqn,ans[1]);
 561 0;
 562 method;
 563 variationofparameters;
 564
 565 /* Kamke 2. 167  */
 566 (pn_(167),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-x*'diff(y,x,1)+(a*x^m+b)*y,y,x));
 567 false;
 568
 569 /* Kamke 2. 168  */
 570 (pn_(168),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1),y,x));
 571 [y=%k2/x+%k1];
 572 ode_check(eqn,ans[1]);
 573 0;
 574 method;
 575 exact;
 576
 577 /* Kamke 2. 169  */
 578 (pn_(169),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1)+(a*x-b^2)*y,y,x));
 579 [y = %k2*bessel_y(sqrt(4*b^2+1),2*sqrt(a)*sqrt(x))/sqrt(x)
 580     +%k1*bessel_j(sqrt(4*b^2+1),2*sqrt(a)*sqrt(x))/sqrt(x)];
 581 [method,ode_check(eqn,ans[1])];
 582 [odelin,0];
 583
 584 /* Kamke 2. 170  */
 585 (pn_(170),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1)+(a*x^2+b)*y,y,x));
 586 [y = %k2*bessel_y(sqrt(1-4*b)/2,-sqrt(a)*x)*sqrt(-x)/x
 587     +%k1*bessel_j(sqrt(1-4*b)/2,-sqrt(a)*x)*sqrt(-x)/x];
 588 [method, ode_check(eqn,ans[1])];
 589 [odelin,0];
 590
 591 /* Kamke 2. 171  */
 592 (pn_(171),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x*'diff(y,x,1)+(L*x^2+a*x-n*(n+1))*y,y,x),0);
 593 0$
 594 /*
 595 [y = %k2*kummer_u(-n-(a*sqrt(-L)/(2*L)),-2*n,-2*sqrt(-L)*x)
 596               *%e^(sqrt(-L)*x)/((-x)^n*x)
 597     +%k1*kummer_m(-n-(a*sqrt(-L)/(2*L)),-2*n,-2*sqrt(-L)*x)
 598               *%e^(sqrt(-L)*x)/((-x)^n*x)];
 599  */
 600 [method, ode_check(eqn,ans[1])];
 601 [odelin,0];
 602
 603 /* Kamke 2. 172  */
 604 (pn_(172),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*(x-1)*'diff(y,x,1)+a*y,y,x),0);
 605 0$
 606 /*
 607 [y = %k2*bessel_y(sqrt(1-4*a)/2,-%i/x)*%e^-(1/x)/sqrt(-x)
 608     +%k1*bessel_j(sqrt(1-4*a)/2,-%i/x)*%e^-(1/x)/sqrt(-x)];
 609  */
 610 [method, ode_check(eqn,ans[1])];
 611 [odelin,0];
 612
 613 /* Kamke 2. 173  */
 614 (pn_(173),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*(x+a)*'diff(y,x,1)-(b-1)*b*y,y,x),0);
 615 0$
 616 /*
 617  [y = %k2*bessel_y(1/2-b,-(%i*a/x))*%e^(a/x)/sqrt(-x)
 618      +%k1*bessel_j(1/2-b,-(%i*a/x))*%e^(a/x)/sqrt(-x)];
 619  */
 620 [method, ode_check(eqn,ans[1])];
 621 [odelin,0];
 622
 623 /* Kamke 2. 174  */
 624 (pn_(174),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+2*y-x^5*log(x),y,x));
 625 [y=(12*x^5*log(x)-7*x^5)/144+%k1*x^2+%k2*x];
 626 ode_check(eqn,ans[1]);
 627 0;
 628 method;
 629 variationofparameters;
 630
 631 /* Kamke 2. 175  */
 632 (pn_(175),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)-4*y-x*sin(x)+(-a*x^2-12*a-4)*cos(x),y,x),0);
 633 0$
 634 /*
 635 [y=-(((2*a+1)*sin(x)+a*x*cos(x))/x)+%k2*x^4-%k1/(5*x)];
 636  */
 637 ode_check(eqn,ans[1]);
 638 0;
 639 method;
 640 variationofparameters;
 641
 642 /* Kamke 2. 176  */
 643 (pn_(176),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+(x^2+2)*y,y,x));
 644 [y = %k2*x*%e^(%i*x)+%k1*x*%e^-(%i*x)];
 645 [method, ode_check(eqn,ans[1])];
 646 [odelin,0];
 647
 648 /* Kamke 2. 177  */
 649 (pn_(177),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x^2+2)*'diff(y,x,1)-2*x*'diff(y,x,1)-x^2/cos(x),y,x));
 650 [y='integrate(%e^-x*(x^2*%e^(2/x)*'integrate(%e^(x-2/x)/(x^2*cos(x)),x)+%k1*x^2*%e^(2/x)),x)+%k2];
 651 ode_check(eqn,ans[1]);
 652 0;
 653 method;
 654 freeofy;
 655
 656 /* Kamke 2. 178  */
 657 (pn_(178),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+(x^2+2)*y-x^3/cos(x),y,x));
 658 [y=x*cos(x)*log(cos(x))+x^2*sin(x)+%k2*x*sin(x)+%k1*x*cos(x)];
 659 [method,ode_check(eqn,ans[1])];
 660 [kovacic,0];
 661
 662 /* Kamke 2. 179  */
 663 (pn_(179),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-2*x*'diff(y,x,1)+(a^2*x^2+2)*y,y,x));
 664 [y = %k2*x*%e^(%i*a*x)+%k1*x*%e^-(%i*a*x)];
 665 [method, ode_check(eqn,ans[1])];
 666 [odelin,0];
 667
 668 /* Kamke 2. 180  */
 669 (pn_(180),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+3*x*'diff(y,x,1)+(x^2-v^2+1)*y-f(x),y,x));
 670 false;
 671
 672 /* Kamke 2. 181  */
 673 (pn_(181),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(3*x-1)*'diff(y,x,1)+y,y,x));
 674 [y = %k2*%e^-(1/x)/x+gamma_incomplete(0,-1/x)*%k1*%e^-(1/x)/x];
 675 ode_check(eqn,ans[1]);
 676 0;
 677 method;
 678 exact;
 679
 680 /* Kamke 2. 182  */
 681 (pn_(182),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-3*x*'diff(y,x,1)+4*y-5*x,y,x));
 682 [y=x^2*(%k2*log(x)+%k1)+5*x];
 683 ode_check(eqn,ans[1]);
 684 0;
 685 method;
 686 variationofparameters;
 687
 688 /* Kamke 2. 183  */
 689 (pn_(183),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-3*x*'diff(y,x,1)-5*y-x^2*log(x),y,x));
 690 [y=-x^2*log(x)/9+%k2*x^5-%k1/(6*x)];
 691 ode_check(eqn,ans[1]);
 692 0;
 693 method;
 694 variationofparameters;
 695
 696 /* Kamke 2. 184  */
 697 (pn_(184),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-4*x*'diff(y,x,1)+6*y-x^4+x^2,y,x));
 698 [y=(2*x^2*log(x)+x^4+2*x^2)/2+%k1*x^3+%k2*x^2];
 699 ode_check(eqn,ans[1]);
 700 0;
 701 method;
 702 variationofparameters;
 703
 704 /* Kamke 2. 185  */
 705 (pn_(185),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+5*x*'diff(y,x,1)+(4-2*x^3)*y,y,x),0);
 706 0$
 707 /*
 708 [y = bessel_y(0,-2*sqrt(2)*sqrt(-x)*x/3)*%k2/x^2
 709     +bessel_j(0,-2*sqrt(2)*sqrt(-x)*x/3)*%k1/x^2];
 710  */
 711 [method,ode_check(eqn,ans[1])];
 712 [odelin,0];
 713
 714 /* Kamke 2. 186  */
 715 (pn_(186),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-5*x*'diff(y,x,1)+8*y-x^3*sin(x),y,x));
 716 [y = (2*x^2*cos(x)+(gamma_incomplete(-1,%i*x)
 717                           +gamma_incomplete(-1,-%i*x))
 718                           *x^4) /4
 719           +%k1*x^4+%k2*x^2];
 720 ode_check(eqn,ans[1]);
 721 0;
 722 method;
 723 variationofparameters;
 724
 725 /* Kamke 2. 187  */
 726 assume(4*b-a^2+2*a-1>0);
 727 [4*b-a^2+2*a-1>0];
 728 (pn_(187),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*x*'diff(y,x,1)+b*y,y,x));
 729 [y=(%k1*sin(sqrt(4*b-a^2+2*a-1)*log(x)/2)+%k2*cos(sqrt(4*b-a^2+2*a-1)*log(x)/2))/x^((a-1)/2)];
 730 ode_check(eqn,ans[1]);
 731 0;
 732 method;
 733 euler;
 734 forget(4*b-a^2+2*a-1>0);
 735 [4*b-a^2+2*a-1>0];
 736
 737 /* Kamke 2. 188  */
 738 (pn_(188),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+c*y,y,x),0);
 739 0$
 740 /*
 741 [y = %k2*kummer_u(-(sqrt(-4*c+a^2-2*a+1)/2)-(a/2)+3/2,
 742                         1-sqrt(-4*c+a^2-2*a+1),-(b/x))
 743               *x^(sqrt(-4*c+a^2-2*a+1)/2-(a/2)+1/2)*%e^(b/x)
 744     +%k1*kummer_m(-(sqrt(-4*c+a^2-2*a+1)/2)-(a/2)+3/2,
 745                         1-sqrt(-4*c+a^2-2*a+1),-(b/x))
 746               *x^(sqrt(-4*c+a^2-2*a+1)/2-(a/2)+1/2)*%e^(b/x)];
 747  */
 748 [method,ode_check(eqn,ans[1])];
 749 [odelin,0];
 750
 751 /* Kamke 2. 189  */
 752 (pn_(189),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+a*x*'diff(y,x,1)+(b*x^m+c)*y,y,x));
 753 false;
 754
 755 /* Kamke 2. 190  */
 756 (pn_(190),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x^2*'diff(y,x,1)+(a*x+b)*y,y,x),0);
 757 0$
 758 /*
 759 [y = %k2*kummer_u(sqrt(1-4*b)/2+a+1/2,sqrt(1-4*b)+1,-x)
 760               *(-x)^(sqrt(1-4*b)/2+1/2)
 761     +%k1*kummer_m(sqrt(1-4*b)/2+a+1/2,sqrt(1-4*b)+1,-x)
 762               *(-x)^(sqrt(1-4*b)/2+1/2)];
 763 */
 764 [method, ode_check(eqn,ans[1])];
 765 [odelin,0];
 766
 767 /* Kamke 2. 191  */
 768 (pn_(191),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x^2*'diff(y,x,1)-2*y,y,x),0);
 769 0$
 770 /*
 771 [y = bessel_y(-3/2,-%i*x/2)*%k2*sqrt(-x)*%e^-(x/2)
 772     +bessel_j(-3/2,-%i*x/2)*%k1*sqrt(-x)*%e^-(x/2)];
 773  */
 774 [method, ode_check(eqn,ans[1])];
 775 [odelin,0];
 776
 777 /* Kamke 2. 192  */
 778 (pn_(192),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+(x^2-1)*'diff(y,x,1)-y,y,x));
 779  [y = %k2*%e^-x*'integrate(%e^(x-1/x),x)+%k1*%e^-x];
 780 [method, ode_check(eqn,ans[1])];
 781 [odelin,0];
 782
 783 /* Kamke 2. 193  */
 784 (pn_(193),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x+1)*'diff(y,x,1)+(x-9)*y,y,x));
 785 [y = kummer_u(-2,-5,-x)*%k2/x^3+kummer_m(-2,-5,-x)*%k1/x^3];
 786 [method, ode_check(eqn,ans[1])];
 787 [odelin,0];
 788
 789 /* Kamke 2. 194  */
 790 (pn_(194),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x+1)*'diff(y,x,1)+(3*x-1)*y,y,x));
 791 [y = kummer_u(4,3,-x)*%k2*x+kummer_m(4,3,-x)*%k1*x];
 792 [method, ode_check(eqn,ans[1])];
 793 [odelin,0];
 794
 795 /* Kamke 2. 195  */
 796 (pn_(195),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+x*(x+3)*'diff(y,x,1)-y,y,x));
 797 [y = kummer_u(sqrt(2)-1,2*sqrt(2)+1,-x)*%k2*x^(sqrt(2)-1)
 798     +kummer_m(sqrt(2)-1,2*sqrt(2)+1,-x)*%k1*x^(sqrt(2)-1)];
 799 [method, ode_check(eqn,ans[1])];
 800 [odelin,0];
 801
 802
 803 /* Kamke 2. 196  */
 804 (pn_(196),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(x-1)*x*'diff(y,x,1)+(x-1)*y,y,x));
 805 [y = gamma_incomplete(-2,-x)*%k2*x+%k1*x];
 806 [method, ode_check(eqn,ans[1])];
 807 [odelin,0];
 808
 809 /* Kamke 2. 197  */
 810 (pn_(197),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(x^2-2*x)*'diff(y,x,1)+(-x-a)*y,y,x),0);
 811 0$
 812 /*
 813 [y = %k2*bessel_y(sqrt(4*a+1)/2,-%i*x/2)*sqrt(-x)*%e^(x/2)/x
 814     +%k1*bessel_j(sqrt(4*a+1)/2,-%i*x/2)*sqrt(-x)*%e^(x/2)/x];
 815  */
 816 [method, ode_check(eqn,ans[1])];
 817 [odelin,0];
 818
 819 /* Kamke 2. 198  */
 820 (pn_(198),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-(x^2-2*x)*'diff(y,x,1)+(-3*x-2)*y,y,x));
 821 [y = gamma_incomplete(-3,x)*%k2*x*%e^x+%k1*x*%e^x];
 822 [method, ode_check(eqn,ans[1])];
 823 [odelin,0];
 824
 825 /* Kamke 2. 199  */
 826 (pn_(199),ans:contrib_ode(eqn:x^2*'diff(y,x,2)-x*(x+4)*'diff(y,x,1)+4*y,y,x));
 827 [y = gamma_incomplete(-3,x)*%k2*x^4*%e^x+%k1*x^4*%e^x];
 828 [method, ode_check(eqn,ans[1])];
 829 [odelin,0];
 830
 831 /* Kamke 2. 200  */
 832 (pn_(200),ans:contrib_ode(eqn:x^2*'diff(y,x,2)+2*x^2*'diff(y,x,1)-(v-1)*v*y,y,x),0);
 833 0$
 834 /*
 835 [y = %k2*bessel_y(1/2-v,-%i*x)*sqrt(-x)*%e^-x
 836     +%k1*bessel_j(1/2-v,-%i*x)*sqrt(-x)*%e^-x];
 837  */
 838 [method, ode_check(eqn,ans[1])];
 839 [odelin,0];