share/contrib/diffequations/tests/rtestode_kamke_2_1.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Second order linear equations 1-100
   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. 1  */
  16 (pn_(1),ans:contrib_ode(eqn:'diff(y,x,2),y,x));
  17 [y=%k2*x+%k1];
  18 ode_check(eqn,ans[1]);
  19 0;
  20 method;
  21 constcoeff;
  22
  23 /* Kamke 2. 2  */
  24 (pn_(2),ans:contrib_ode(eqn:'diff(y,x,2)+y,y,x));
  25 [y=%k1*sin(x)+%k2*cos(x)];
  26 ode_check(eqn,ans[1]);
  27 0;
  28 method;
  29 constcoeff;
  30
  31 /* Kamke 2. 3  */
  32 assume(not(equal(n,1)));
  33 [notequal(n,1)];
  34 (pn_(3),ans:contrib_ode(eqn:'diff(y,x,2)+y-sin(n*x),y,x));
  35 [y=-sin(n*x)/(n^2-1)+%k1*sin(x)+%k2*cos(x)];
  36 ode_check(eqn,ans[1]);
  37 0;
  38 method;
  39 variationofparameters;
  40 forget(not(equal(n,1)));
  41 [notequal(n,1)];
  42
  43 /* Kamke 2. 4  */
  44 assume(not(equal(b,1)));
  45 [notequal(b,1)];
  46 (pn_(4),ans:contrib_ode(eqn:'diff(y,x,2)+y-a*cos(b*x),y,x));
  47 [y=-a*cos(b*x)/(b^2-1)+%k1*sin(x)+%k2*cos(x)];
  48 ode_check(eqn,ans[1]);
  49 0;
  50 method;
  51 variationofparameters;
  52 forget(not(equal(b,1)));
  53 [notequal(b,1)];
  54
  55 /* Kamke 2. 5  */
  56 (pn_(5),ans:contrib_ode(eqn:'diff(y,x,2)+y-sin(a*x)*sin(b*x),y,x));
  57 [y=((b^2-2*a*b+a^2-1)*cos((b+a)*x)+(-b^2-2*a*b-a^2+1)*cos((b-a)*x))/(2*b^4+(-4*a^2-4)*b^2+2*a^4-4*a^2+2)+%k1*sin(x)+%k2*cos(x)];
  58 [method,ode_check(eqn,ans[1])];
  59 [variationofparameters,0];
  60
  61 /* Kamke 2. 6  */
  62 (pn_(6),ans:contrib_ode(eqn:'diff(y,x,2)-y,y,x));
  63 [y=%k1*%e^x+%k2*%e^-x];
  64 ode_check(eqn,ans[1]);
  65 0;
  66 method;
  67 constcoeff;
  68
  69 /* Kamke 2. 7  */
  70 (pn_(7),ans:contrib_ode(eqn:'diff(y,x,2)-2*y-4*x^2*%e^x^2,y,x));
  71 [y=%e^x^2+%k1*%e^(2^(1/2)*x)+%k2*%e^-(2^(1/2)*x)];
  72 ode_check(eqn,ans[1]);
  73 0;
  74 method;
  75 variationofparameters;
  76
  77 /* Kamke 2. 8  CHECKME */
  78 assume(a>0);
  79 [a>0];
  80 (pn_(8),ans:contrib_ode(eqn:'diff(y,x,2)+a^2*y-cot(a*x),y,x));
  81 [y = -((sin(a*x)*log(cos(a*x)+1)-sin(a*x)*log(cos(a*x)-1))/(2*a^2))
  82    +%k1*sin(a*x)+%k2*cos(a*x)];
  83 [method,ode_check(eqn,ans[1])];
  84 [variationofparameters,0];
  85 forget(a>0);
  86 [a>0];
  87
  88 /* Kamke 2. 9  */
  89 assume(l>0);
  90 [l>0];
  91 (pn_(9),ans:contrib_ode(eqn:'diff(y,x,2)+l*y,y,x));
  92 [y=%k1*sin(sqrt(l)*x)+%k2*cos(sqrt(l)*x)];
  93 ode_check(eqn,ans[1]);
  94 0;
  95 method;
  96 constcoeff;
  97 forget(l>0);
  98 [l>0];
  99
 100 /* Kamke 2. 10  */
 101 (pn_(10),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*y,y,x));
 102 [y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*abs(a)))*%k2*sqrt(a*x+b)
 103    + bessel_j(1/3,2*(a*x+b)^(3/2)/(3*abs(a)))*%k1*sqrt(a*x+b)];
 104 [method,ode_check(eqn,ans[1])];
 105 [odelin,0];
 106
 107 /* Kamke 2. 11  */
 108 (pn_(11),ans:contrib_ode(eqn:'diff(y,x,2)-(x^2+1)*y,y,x));
 109 [y = %k2*%e^(x^2/2)*erf(x)+%k1*%e^(x^2/2)];
 110 [method,ode_check(eqn,ans[1])];
 111 [odelin,0];
 112
 113 /* Kamke 2. 12  */
 114 (pn_(12),ans:contrib_ode(eqn:'diff(y,x,2)-(x^2+a)*y,y,x),0);
 115 0$
 116 /*
 117 [y = (%k2*kummer_u(-(a-1)/4,1/2,-x^2)*%e^(x^2/2)*sqrt(abs(x)))/sqrt(-x)
 118    + (%k1*kummer_m(-(a-1)/4,1/2,-x^2)*%e^(x^2/2)*sqrt(abs(x)))/sqrt(-x)];
 119   */
 120 [method,ode_check(eqn,ans[1])];
 121 [odelin,0];
 122
 123 /* Kamke 2. 13  */
 124 (pn_(13),ans:contrib_ode(eqn:'diff(y,x,2)-(a^2*x^2+a)*y,y,x));
 125 [y = %k2*%e^(a*x^2/2)*erf(sqrt(a)*x)+%k1*%e^(a*x^2/2)];
 126 [method,ode_check(eqn,ans[1])];
 127 [odelin,0];
 128
 129 /* Kamke 2. 14  */
 130 (pn_(14),ans:contrib_ode(eqn:'diff(y,x,2)-c*x^a*y,y,x));
 131 false;
 132
 133 /* Kamke 2. 15  */
 134 (pn_(15),ans:contrib_ode(eqn:'diff(y,x,2)-(a^2*x^(2*n)-1)*y,y,x));
 135 false;
 136
 137 /* Kamke 2. 16  */
 138 (pn_(16),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x^(2*c)+b*x^(c-1))*y,y,x));
 139 false;
 140
 141 /* Kamke 2. 17  */
 142 (pn_(17),ans:contrib_ode(eqn:'diff(y,x,2)+(%e^(2*x)-v^2)*y,y,x));
 143 false;
 144
 145 /* Kamke 2. 18  */
 146 (pn_(18),ans:contrib_ode(eqn:'diff(y,x,2)+a*%e^(b*x)*y,y,x));
 147 false;
 148
 149 /* Kamke 2. 19  */
 150 (pn_(19),ans:contrib_ode(eqn:'diff(y,x,2)-(4*a^2*b^2*x^2*%e^(2*b*x^2)-1)*y,y,x));
 151 false;
 152
 153 /* Kamke 2. 20  */
 154 (pn_(20),ans:contrib_ode(eqn:'diff(y,x,2)+(a*%e^(2*x)+b*%e^x+c)*y,y,x));
 155 false;
 156
 157 /* Kamke 2. 21  */
 158 (pn_(21),ans:contrib_ode(eqn:'diff(y,x,2)+(a*cos(x)^2+b)*y,y,x));
 159 false;
 160
 161 /* Kamke 2. 22  */
 162 (pn_(22),ans:contrib_ode(eqn:'diff(y,x,2)+(a*cos(2*x)+b)*y,y,x));
 163 false;
 164
 165 /* Kamke 2. 23  */
 166 (pn_(23),ans:contrib_ode(eqn:'diff(y,x,2)+(a*cos(x)^2+b)*y,y,x));
 167 false;
 168
 169 /* Kamke 2. 24  */
 170 (pn_(24),ans:contrib_ode(eqn:'diff(y,x,2)-(2*tan(x)^2+1)*y,y,x));
 171 false;
 172
 173 /* Kamke 2. 25  */
 174 (pn_(25),ans:contrib_ode(eqn:'diff(y,x,2)-((n-1)*n/sin(x)^2+(m-1)*m/cos(x)^2+a)*y,y,x));
 175 false;
 176
 177 /* Kamke 2. 26  */
 178 (pn_(26),ans:contrib_ode(eqn:'diff(y,x,2)-y*(B+p(x)*A),y,x));
 179 false;
 180
 181 /* Kamke 2. 27  */
 182 (pn_(27),ans:contrib_ode(eqn:'diff(y,x,2)+(a*jacobi_sn(x,m)^2+b)*y,y,x));
 183 false;
 184
 185 /* Kamke 2. 28  */
 186 (pn_(28),ans:contrib_ode(eqn:'diff(y,x,2)-('diff(p(x),x,4)/30+7*'diff(p(x),x,2)/3+a*p(x)+b)*y,y,x));
 187 false;
 188
 189 /* Kamke 2. 29  */
 190 (pn_(29),ans:contrib_ode(eqn:'diff(y,x,2)-('diff(f(x),x,1)+f(x)^2)*y,y,x));
 191 false;
 192
 193 /* Kamke 2. 30  */
 194 (pn_(30),ans:contrib_ode(eqn:'diff(y,x,2)+(P(x)+l)*y,y,x));
 195 false;
 196
 197 /* Kamke 2. 31  */
 198 (pn_(31),ans:contrib_ode(eqn:'diff(y,x,2)-f(x)*y,y,x));
 199 false;
 200
 201 /* Kamke 2. 32  */
 202 (pn_(32),ans:contrib_ode(eqn:'diff(y,x,2)+('diff(g(x),x,3)/(2*'diff(g(x),x,1))-3*('diff(g(x),x,2))^2/(4*('diff(g(x),x,1))^2)+(1/4-v^2)*('diff(g(x),x,1))^2/g(x)+('diff(g(x),x,1))^2)*y,y,x));
 203 false;
 204
 205 /* Kamke 2. 33  */
 206 (pn_(33),ans:contrib_ode(eqn:'diff(y,x,2)+'diff(y,x,1)+a*%e^-(2*x)*y,y,x));
 207 [y=%k2*cos(sqrt(a)*%e^-x)-%k1*sin(sqrt(a)*%e^-x)];
 208 ode_check(eqn,ans[1]);
 209 0;
 210 method;
 211 xformtoconstcoeff;
 212
 213 /* Kamke 2. 34  */
 214 (pn_(34),ans:contrib_ode(eqn:'diff(y,x,2)-'diff(y,x,1)+%e^(2*x)*y,y,x));
 215 [y=%k1*sin(%e^x)+%k2*cos(%e^x)];
 216 ode_check(eqn,ans[1]);
 217 0;
 218 method;
 219 xformtoconstcoeff;
 220
 221 /* Kamke 2. 35  */
 222 assume(a^2>4*b);
 223 [a^2>4*b];
 224 (pn_(35),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(y,x,1)+b*y,y,x));
 225 [y=%k1*%e^((sqrt(a^2-4*b)-a)*x/2)+%k2*%e^((-sqrt(a^2-4*b)-a)*x/2)];
 226 ode_check(eqn,ans[1]);
 227 0;
 228 method;
 229 constcoeff;
 230 forget(a^2>4*b);
 231 [a^2>4*b];
 232
 233 /* Kamke 2. 36  */
 234 assume(a^2>4*b);
 235 [a^2>4*b];
 236 (pn_(36),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(y,x,1)+b*y-f(x),y,x),0);
 237 /*
 238 [y=-%e^(-sqrt(a^2-4*b)*x/2-a*x/2)*('integrate(%e^(sqrt(a^2-4*b)*x/2+a*x/2)*f(x),x)-%e^(sqrt(a^2-4*b)*x)*'integrate(%e^(a*x/2-sqrt(a^2-4*b)*x/2)*f(x),x))/sqrt(a^2-4*b)+%k1*%e^((sqrt(a^2-4*b)-a)*x/2)+%k2*%e^((-sqrt(a^2-4*b)-a)*x/2)];
 239  */
 240 0$
 241 ode_check(eqn,ans[1]);
 242 0$
 243 method;
 244 variationofparameters;
 245 forget(a^2>4*b);
 246 [a^2>4*b];
 247
 248 /* Kamke 2. 37  */
 249 (pn_(37),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(y,x,1)+(-b^2*x^2-c)*y,y,x),0);
 250 0$
 251 /*
 252 [y = (%k2*kummer_u(-(4*c-4*b+a^2)/(16*b),1/2,-b*x^2)
 253                *%e^((b*x^2)/2-(a*x)/2)*sqrt(abs(x)))/sqrt(-x)
 254     +(%k1*kummer_m(-(4*c-4*b+a^2)/(16*b),1/2,-b*x^2)
 255                *%e^((b*x^2)/2-(a*x)/2)*sqrt(abs(x)))/sqrt(-x)]$
 256  */
 257 [method,ode_check(eqn,expand(ans[1]))];
 258 [odelin,0];
 259
 260 /* Kamke 2. 38  */
 261 (pn_(38),ans:contrib_ode(eqn:'diff(y,x,2)+2*a*'diff(y,x,1)+f(x)*y,y,x));
 262 false;
 263
 264 /* Kamke 2. 39  */
 265 (pn_(39),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)+y,y,x));
 266 [y=%k2*%e^-(x^2/2)-sqrt(2)*sqrt(%pi)*%i*%k1*%e^-(x^2/2)*erf(%i*x/sqrt(2))/2];
 267 ode_check(eqn,ans[1]);
 268 0;
 269 method;
 270 exact;
 271
 272 /* Kamke 2. 40  */
 273 (pn_(40),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)-y,y,x),0);
 274 0$
 275 /*
 276 [y = kummer_u(0,3/2,-x^2/2)*%k2*x+kummer_m(0,3/2,-x^2/2)*%k1*x];
 277  */
 278 [method,ode_check(eqn,ans[1])];
 279 [odelin,0];
 280
 281 /* Kamke 2. 41  */
 282 (pn_(41),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)+(n+1)*y,y,x),0);
 283 0$
 284 /*
 285 [y = %k2*kummer_u(n/2+1/2,1/2,-(x^2/2))
 286     +%k1*kummer_m(n/2+1/2,1/2,-(x^2/2))];
 287  */
 288 [method,ode_check(eqn,ans[1])];
 289 [odelin,0];
 290
 291 /* Kamke 2. 42  */
 292 (pn_(42),ans:contrib_ode(eqn:'diff(y,x,2)+x*'diff(y,x,1)-n*y,y,x),0);
 293 0$
 294 /*
 295 [y = %k2*kummer_u(-n/2,1/2,-x^2/2)
 296     +%k1*kummer_m(-n/2,1/2,-x^2/2)];
 297  */
 298 [method,ode_check(eqn,ans[1])];
 299 [odelin,0];
 300
 301 /* Kamke 2. 43  */
 302 (pn_(43),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)+2*y,y,x),0);
 303 0$
 304 /*
 305 [y = kummer_u(2,3/2,-x^2/2)*%k2*x*%e^(x^2/2)
 306     +kummer_m(2,3/2,-x^2/2)*%k1*x*%e^(x^2/2)];
 307  */
 308 [method,ode_check(eqn,ans[1])];
 309 [odelin,0];
 310
 311 /* Kamke 2. 44  */
 312 (pn_(44),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)-a*y,y,x),0);
 313 0$
 314 /*
 315 [y = %k2*kummer_u(1/2-(a/2),1/2,-(x^2/2))*%e^(x^2/2)
 316     +%k1*kummer_m(1/2-(a/2),1/2,-(x^2/2))*%e^(x^2/2)];
 317  */
 318 [method,ode_check(eqn,ans[1])];
 319 [odelin,0];
 320
 321 /* Kamke 2. 45  */
 322 (pn_(45),ans:contrib_ode(eqn:'diff(y,x,2)-x*'diff(y,x,1)+(x-1)*y,y,x),0);
 323 0$
 324 /*
 325 [y = %k2*%e^(x-2)*erf(%i*x/sqrt(2)-2*%i/sqrt(2))+%k1*%e^x];
 326  */
 327 [method,ode_check(eqn,ans[1])];
 328 [odelin,0];
 329
 330 /* Kamke 2. 46  */
 331 (pn_(46),ans:contrib_ode(eqn:'diff(y,x,2)-2*x*'diff(y,x,1)+a*y,y,x),0);
 332 0$
 333 /*
 334 [y = %k2*kummer_u(a/4+1/2,1/2,-x^2)*%e^x^2
 335     +%k1*kummer_m(a/4+1/2,1/2,-x^2)*%e^x^2];
 336  */
 337 [method,ode_check(eqn,ans[1])];
 338 [odelin,0];
 339
 340 /* Kamke 2. 47  */
 341 (pn_(47),ans:contrib_ode(eqn:'diff(y,x,2)+4*x*'diff(y,x,1)+(4*x^2+2)*y,y,x));
 342 [y = %k2*x*%e^-x^2+%k1*%e^-x^2];
 343 [method,ode_check(eqn,ans[1])];
 344 [odelin,0];
 345
 346 /* Kamke 2. 48  */
 347 (pn_(48),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(3*x^2+2*n-1)*y,y,x),0);
 348 0$
 349 /*
 350 [y = %k2*kummer_u(n/2+1/2,1/2,-x^2)*%e^(3*x^2/2)
 351     +%k1*kummer_m(n/2+1/2,1/2,-x^2)*%e^(3*x^2/2)];
 352  */
 353 [method,ode_check(eqn,ans[1])];
 354 [odelin,0];
 355
 356 /* Kamke 2. 49  */
 357 (pn_(49),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(4*x^2-1)*y-%e^x,y,x));
 358 [y = (sqrt(%pi)*((%e^x^2*sin((2*x-1)/2)-%i*%e^x^2*cos((2*x-1)/2))
 359                       *erf((2*x+%i-1)/2)
 360                       +(%e^x^2*sin((2*x-1)/2)+%i*%e^x^2*cos((2*x-1)/2))
 361                        *erf((2*x-%i-1)/2)))
 362          /4
 363          +%k2*%e^x^2*sin(x)+%k1*%e^x^2*cos(x)];
 364 [method,ode_check(eqn,ans[1])];
 365 [kovacic,0];
 366
 367 /* Kamke 2. 50  */
 368 (pn_(50),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(4*x^2-2)*y,y,x));
 369 [y = %k2*x*%e^x^2+%k1*%e^x^2];
 370 [method,ode_check(eqn,ans[1])];
 371 [odelin,0];
 372
 373 /* Kamke 2. 51  */
 374 (pn_(51),ans:contrib_ode(eqn:'diff(y,x,2)-4*x*'diff(y,x,1)+(4*x^2-3)*y-%e^x^2,y,x));
 375 [y = %k2*%e^(x^2+x)+%k1*%e^(x^2-x)-%e^x^2];
 376 [method,ode_check(eqn,ans[1])];
 377 [kovacic,0];
 378
 379 /* Kamke 2. 52  */
 380 (pn_(52),ans:contrib_ode(eqn:'diff(y,x,2)+a*x*'diff(y,x,1)+b*y,y,x),0);
 381 0$
 382 /*
 383  [y = %k2*kummer_u(b/(2*a),1/2,-a*x^2/2)
 384      +%k1*kummer_m(b/(2*a),1/2,-a*x^2/2)];
 385  */
 386 [method,ode_check(eqn,ans[1])];
 387 [odelin,0];
 388
 389 /* Kamke 2. 53  */
 390 (pn_(53),ans:contrib_ode(eqn:'diff(y,x,2)+2*a*x*'diff(y,x,1)+a^2*x^2*y,y,x));
 391 [y = %k2*%e^(sqrt(a)*x-(a*x^2/2))+%k1*%e^(-(a*x^2/2)-sqrt(a)*x)];
 392 [method,ode_check(eqn,ans[1])];
 393 [odelin,0];
 394
 395 /* Kamke 2. 54  */
 396 /* CHECKME: Takes a long time
 397 (pn_(54),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+(c*x+d)*y,y,x));
 398 false;
 399 */
 400
 401 /* Kamke 2. 55  */
 402 /* CHECKME: Takes a long time
 403 (pn_(55),ans:contrib_ode(eqn:'diff(y,x,2)+(a*x+b)*'diff(y,x,1)+(a1*x^2+b1*x+c_1)*y,y,x));
 404 false;
 405 */
 406
 407 /* Kamke 2. 56  */
 408 (pn_(56),ans:contrib_ode(eqn:'diff(y,x,2)-x^2*'diff(y,x,1)+x*y,y,x),0);
 409 0$
 410 /*
 411 [y = kummer_u(1,2/3,-x^3/3)*%k2*%e^(x^3/3)
 412     +kummer_m(1,2/3,-x^3/3)*%k1*%e^(x^3/3)];
 413  */
 414 [method, ode_check(eqn,ans[1])];
 415 [odelin,0];
 416
 417 /* Kamke 2. 57  */
 418 (pn_(57),ans:contrib_ode(eqn:'diff(y,x,2)-x^2*'diff(y,x,1)-(x+1)^2*y,y,x));
 419 [y=%k2*%e^(x^3/3+x)*'integrate(%e^(-(x^3/3)-2*x),x)+%k1*%e^(x^3/3+x)];
 420 [method, ode_check(eqn,ans[1])];
 421 [odelin,0];
 422
 423 /* Kamke 2. 58  */
 424 (pn_(58),ans:contrib_ode(eqn:'diff(y,x,2)-x^2*(x+1)*'diff(y,x,1)+x*(x^4-2)*y,y,x));
 425 [y = %k2*%e^(x^3/3)*'integrate(%e^(x^4/4-x^3/3),x)+%k1*%e^(x^3/3)];
 426 [method, ode_check(eqn,ans[1])];
 427 [odelin,0];
 428
 429 /* Kamke 2. 59  */
 430 (pn_(59),ans:contrib_ode(eqn:'diff(y,x,2)+x^4*'diff(y,x,1)-x^3*y,y,x));
 431 [y = kummer_u(0,6/5,-x^5/5)*%k2*x+kummer_m(0,6/5,-x^5/5)*%k1*x];
 432 [method, ode_check(eqn,ans[1])];
 433 [odelin,0];
 434
 435 /* Kamke 2. 60  */
 436 /* CHECKME: Takes a long time
 437 (pn_(60),ans:contrib_ode(eqn:'diff(y,x,2)+a*x^(q-1)*'diff(y,x,1)+b*x^(q-2)*y,y,x));
 438 false;
 439 */
 440
 441 /* Kamke 2. 61  */
 442 (pn_(61),ans:contrib_ode(eqn:'diff(y,x,2)+sqrt(x)*'diff(y,x,1)+(x/4+1/(4*sqrt(x))-9)*y-x*%e^-(x^(3/2)/3),y,x));
 443 [y=(-(x*%e^-(x^(3/2)/3))/9)+%k2*%e^(3*x-x^(3/2)/3)+%k1*%e^((-x^(3/2)/3)-3*x)];
 444 [method,ode_check(eqn,ans[1])];
 445 [kovacic,0];
 446
 447 /* Kamke 2. 62  */
 448 (pn_(62),ans:contrib_ode(eqn:'diff(y,x,2)-'diff(y,x,1)/sqrt(x)+(x+sqrt(x)-8)*y/(4*x^2),y,x));
 449 false;
 450
 451 /* Kamke 2. 63  */
 452 (pn_(63),ans:contrib_ode(eqn:'diff(y,x,2)-(2*%e^x+1)*'diff(y,x,1)+%e^(2*x)*y-%e^(3*x),y,x));
 453 [y=(%k2*%e^x+%k1)*%e^%e^x+%e^x+2];
 454 ode_check(eqn,ans[1]);
 455 0;
 456 method;
 457 variationofparameters;
 458
 459 /* Kamke 2. 64  */
 460 assume(a^2>4*b);
 461 [a^2>4*b];
 462 (pn_(64),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(y,x,1)+b*y+tan(x),y,x),0);
 463 0$
 464 /*
 465 [y=%e^(-sqrt(a^2-4*b)*x/2-a*x/2)*('integrate(%e^(sqrt(a^2-4*b)*x/2+a*x/2)*tan(x),x)-%e^(sqrt(a^2-4*b)*x)*'integrate(%e^(a*x/2-sqrt(a^2-4*b)*x/2)*tan(x),x))/sqrt(a^2-4*b)+%k1*%e^((sqrt(a^2-4*b)-a)*x/2)+%k2*%e^((-sqrt(a^2-4*b)-a)*x/2)];
 466  */
 467 ode_check(eqn,ans[1]);
 468 0;
 469 method;
 470 variationofparameters;
 471 forget(a^2>4*b);
 472 [a^2>4*b];
 473
 474 /* Kamke 2. 65  */
 475 (pn_(65),ans:contrib_ode(eqn:'diff(y,x,2)+2*n*cot(x)*'diff(y,x,1)+(n^2-a^2)*y,y,x));
 476 false;
 477
 478 /* Kamke 2. 66  */
 479 (pn_(66),ans:contrib_ode(eqn:'diff(y,x,2)+tan(x)*'diff(y,x,1)+cos(x)^2*y,y,x));
 480 false;
 481
 482 /* Kamke 2. 67  */
 483 (pn_(67),ans:contrib_ode(eqn:'diff(y,x,2)+tan(x)*'diff(y,x,1)-cos(x)^2*y,y,x));
 484 false;
 485
 486 /* Kamke 2. 68  */
 487 (pn_(68),ans:contrib_ode(eqn:'diff(y,x,2)+cot(x)*'diff(y,x,1)+v*(v+1)*y,y,x));
 488 false;
 489
 490 /* Kamke 2. 69  */
 491 (pn_(69),ans:contrib_ode(eqn:'diff(y,x,2)-cot(x)*'diff(y,x,1)+sin(x)^2*y,y,x));
 492 false;
 493
 494 /* Kamke 2. 70  */
 495 (pn_(70),ans:contrib_ode(eqn:'diff(y,x,2)+a*tan(x)*'diff(y,x,1)+b*y,y,x));
 496 false;
 497
 498 /* Kamke 2. 71  */
 499 (pn_(71),ans:contrib_ode(eqn:'diff(y,x,2)+2*a*cot(a*x)*'diff(y,x,1)+(b^2-a^2)*y,y,x));
 500 false;
 501
 502 /* Kamke 2. 72  */
 503 (pn_(72),ans:contrib_ode(eqn:'diff(y,x,2)+a*'diff(p(x),x,2)*'diff(y,x,1)+(-4*a*n*p(x)^2+b*p(x)+a)*y,y,x));
 504 false;
 505
 506 /* Kamke 2. 73  */
 507 (pn_(73),ans:contrib_ode(eqn:'diff(y,x,2)+('diff(p(x),x,3)-'diff(p(x),x,2)-p(x)*'diff(p(x),x,1))*'diff(y,x,1)/('diff(p(x),x,1)+p(x)^2)+(-p(x)*'diff(p(x),x,2)+('diff(p(x),x,1))^2-p(x)^2*'diff(p(x),x,1))*y/('diff(p(x),x,1)+p(x)^2),y,x));
 508 false;
 509
 510 /* Kamke 2. 74  */
 511 (pn_(74),ans:contrib_ode(eqn:'diff(y,x,2)+m*jacobi_cn(x,m)*jacobi_sn(x,m)*'diff(y,x,1)/jacobi_dn(x,m)+n^2*jacobi_dn(x,m)^2*y,y,x));
 512 [y = %k1*sin(n*asin(jacobi_sn(x,m)))+%k2*cos(n*asin(jacobi_sn(x,m)))];
 513 method;
 514 xformtoconstcoeff;
 515 /* solution satisfies the ODE for a few random values of the variables */
 516
 517 /* Kamke 2. 75  */
 518 (pn_(75),ans:contrib_ode(eqn:'diff(y,x,2)+f(x)*'diff(y,x,1)+g(x)*y,y,x));
 519 false;
 520
 521 /* Kamke 2. 76  */
 522 (pn_(76),ans:contrib_ode(eqn:'diff(y,x,2)+f(x)*'diff(y,x,1)+('diff(f(x),x,1)+a)*y-g(x),y,x));
 523 false;
 524
 525 /* Kamke 2. 77  */
 526 (pn_(77),ans:contrib_ode(eqn:'diff(y,x,2)+(a*f(x)+b)*'diff(y,x,1)+(c*f(x)+d)*y,y,x));
 527 false;
 528
 529 /* Kamke 2. 78  */
 530 (pn_(78),ans:contrib_ode(eqn:'diff(y,x,2)+f(x)*'diff(y,x,1)+('diff(f(x),x,1)/2+f(x)^2/4+a)*y,y,x));
 531 false;
 532
 533 /* Kamke 2. 79  */
 534 (pn_(79),ans:contrib_ode(eqn:'diff(y,x,2)-a*'diff(f(x),x,1)*'diff(y,x,1)/f(x)+b*f(x)^(2*a)*y,y,x));
 535 [y=%k1*sin(sqrt(b)*'integrate(f(x)^a,x))+%k2*cos(sqrt(b)*'integrate(f(x)^a,x))];
 536 ode_check(eqn,ans[1]);
 537 0;
 538 method;
 539 xformtoconstcoeff;
 540
 541 /* Kamke 2. 80  */
 542 (pn_(80),ans:contrib_ode(eqn:'diff(y,x,2)-('diff(f(x),x,1)/f(x)+2*a)*'diff(y,x,1)+(a*'diff(f(x),x,1)/f(x)-b^2*f(x)^2+a^2)*y,y,x));
 543 false;
 544
 545 /* Kamke 2. 81  */
 546 (pn_(81),ans:contrib_ode(eqn:'diff(y,x,2)+f(x)*'diff(f(x),x,3)*'diff(y,x,1)/(f(x)^2+b^2)-a^2*('diff(f(x),x,1))^2*y/(f(x)^2+b^2),y,x));
 547 false;
 548
 549 /* Kamke 2. 82  */
 550 (pn_(82),ans:contrib_ode(eqn:'diff(y,x,2)-('diff(g(x),x,2)/'diff(g(x),x,1)+(2*m-1)*'diff(g(x),x,1)/g(x))*'diff(y,x,1)+((m^2-v^2)*('diff(g(x),x,1))^2/g(x)+('diff(g(x),x,1))^2)*y,y,x));
 551 false;
 552
 553 /* Kamke 2. 83  */
 554 (pn_(83),ans:contrib_ode(eqn:'diff(y,x,2)-'diff(f(x),x,1)*'diff(y,x,1)/f(x)+('diff(g(x),x,3)/(2*'diff(g(x),x,1))-3*('diff(g(x),x,2))^2/(4*('diff(g(x),x,1))^2)+(1/4-v^2)*('diff(g(x),x,1))^2/g(x)^2+('diff(g(x),x,1))^2-'diff(f(x),x,2)/(2*f(x))+3*('diff(f(x),x,1))^2/(4*f(x)^2))*y,y,x));
 555 false;
 556
 557 /* Kamke 2. 84  */
 558 (pn_(84),ans:contrib_ode(eqn:'diff(y,x,2)-('diff(g(x),x,2)/'diff(g(x),x,1)-'diff(g(x),x,1)/g(x)+2*'diff(f(x),x,1)/f(x))*'diff(y,x,1)+('diff(f(x),x,1)*('diff(g(x),x,2)/'diff(g(x),x,1)-'diff(g(x),x,1)/g(x)+2*'diff(f(x),x,1)/f(x))/f(x)-v^2*('diff(g(x),x,1))^2/g(x)^2+('diff(g(x),x,1))^2-'diff(f(x),x,2)/f(x))*y,y,x));
 559 false;
 560
 561 /* Kamke 2. 85  */
 562 (pn_(85),ans:contrib_ode(eqn:'diff(y,x,2)-(2*'diff(h(x),x,1)/h(x)+'diff(g(x),x,2)/'diff(g(x),x,1)+(2*v-1)*'diff(g(x),x,1)/g(x))*'diff(y,x,1)+(-'diff(h(x),x,2)/h(x)+'diff(h(x),x,1)*(2*'diff(h(x),x,1)/h(x)+'diff(g(x),x,2)/'diff(g(x),x,1)+(2*v-1)*'diff(g(x),x,1)/g(x))/h(x)+('diff(g(x),x,1))^2)*y,y,x));
 563 false;
 564
 565 /* Kamke 2. 86  */
 566 (pn_(86),ans:contrib_ode(eqn:4*'diff(y,x,2)+9*x*y,y,x),0);
 567 0$
 568 /*
 569 [y = bessel_y(1/3,x^(3/2))*%k2*sqrt(x)+bessel_j(1/3,x^(3/2))*%k1*sqrt(x)];
 570  */
 571 [method,ode_check(eqn,ans[1])];
 572 [odelin,0];
 573
 574 /* Kamke 2. 87  */
 575 (pn_(87),ans:contrib_ode(eqn:4*'diff(y,x,2)-(x^2+a)*y,y,x),0);
 576 0$
 577 /*
 578 [y = %k2*kummer_u(1/4-(a/8),1/2,-(x^2/2))*%e^(x^2/4)
 579     +%k1*kummer_m(1/4-(a/8),1/2,-(x^2/2))*%e^(x^2/4)];
 580  */
 581 [method, ode_check(eqn,ans[1])];
 582 [odelin,0];
 583
 584 /* Kamke 2. 88  */
 585 (pn_(88),ans:contrib_ode(eqn:4*'diff(y,x,2)+4*tan(x)*'diff(y,x,1)+(-5*tan(x)^2-2)*y,y,x));
 586 false;
 587
 588 /* Kamke 2. 89  */
 589 (pn_(89),ans:contrib_ode(eqn:a*'diff(y,x,2)-(x+c+a*b)*'diff(y,x,1)+(b*(x+c)+d)*y,y,x),0);
 590 0$
 591 /*
 592 [y=%k2*kummer_u(d/2+1/2,1/2,-(x^2/(2*a))-(c*x/a)+b*x-(c^2/(2*a))+b*c-(a*b^2/2))
 593               *%e^(x^2/(2*a)+c*x/a+c^2/(4*a)-(b*c/2)+a*b^2/4)
 594   +%k1*kummer_m(d/2+1/2,1/2,-(x^2/(2*a))-(c*x/a)+b*x-(c^2/(2*a))+b*c-(a*b^2/2))
 595              *%e^(x^2/(2*a)+c*x/a+c^2/(4*a)-(b*c/2)+a*b^2/4)];
 596  */
 597 [method, ode_check(eqn,ans[1])];
 598 [odelin,0];
 599
 600 /* Kamke 2. 90  */
 601 (pn_(90),ans:contrib_ode(eqn:a^2*'diff(y,x,2)+a*(a^2-2*b*%e^-(a*x))*'diff(y,x,1)+b^2*%e^-(2*a*x)*y,y,x));
 602 [y=(%k1-%k2*b*%e^-(a*x)/a^2)*%e^-(b*%e^-(a*x)/a^2)];
 603 ode_check(eqn,ans[1]);
 604 0;
 605 method;
 606 xformtoconstcoeff;
 607
 608 /* Kamke 2. 91  */
 609 (pn_(91),ans:contrib_ode(eqn:x*('diff(y,x,2)+y)-cos(x),y,x));
 610 [y = ((2*log(x)-gamma_incomplete(0,2*%i*x)-gamma_incomplete(0,-2*%i*x))*sin(x)
 611     +(%i*gamma_incomplete(0,2*%i*x)-%i*gamma_incomplete(0,-2*%i*x))*cos(x))
 612          /4 + %k1*sin(x)+%k2*cos(x)]$
 613 [method,ode_check(eqn,ans[1])];
 614 [variationofparameters,0];
 615
 616 /* Kamke 2. 92  */
 617 (pn_(92),ans:contrib_ode(eqn:x*'diff(y,x,2)+(x+a)*y,y,x));
 618 [y = %k2*kummer_u(-%i*a/2,0,-2*%i*x)*%e^(%i*x)
 619     +%k1*kummer_m(-%i*a/2,0,-2*%i*x)*%e^(%i*x)];
 620 [method, ode_check(eqn,ans[1])];
 621 [odelin,0];
 622
 623 /* Kamke 2. 93  */
 624 (pn_(93),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1),y,x));
 625 [y=%k1*log(x)+%k2];
 626 ode_check(eqn,ans[1]);
 627 0;
 628 method;
 629 exact;
 630
 631 /* Kamke 2. 94  */
 632 (pn_(94),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+a*y,y,x));
 633 [y = bessel_y(0,2*sqrt(a)*sqrt(x))*%k2
 634     +bessel_j(0,2*sqrt(a)*sqrt(x))*%k1];
 635 [method,ode_check(eqn,ans[1])];
 636 [odelin,0];
 637
 638 /* Kamke 2. 95  */
 639 (pn_(95),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+l*x*y,y,x));
 640 [y = bessel_y(0,-sqrt(l)*x)*%k2+bessel_j(0,-sqrt(l)*x)*%k1];
 641 [method,ode_check(eqn,ans[1])];
 642 [odelin,0];
 643
 644 /* Kamke 2. 96  */
 645 (pn_(96),ans:contrib_ode(eqn:x*'diff(y,x,2)+'diff(y,x,1)+(x+a)*y,y,x),0);
 646 0$
 647 /*
 648 [y = %k2*kummer_u(1/2-((%i*a)/2),1,-2*%i*x)*%e^(%i*x)
 649     +%k1*kummer_m(1/2-((%i*a)/2),1,-2*%i*x)*%e^(%i*x)];
 650  */
 651 [method, ode_check(eqn,ans[1])];
 652 [odelin,0];
 653
 654 /* Kamke 2. 97  */
 655 (pn_(97),ans:contrib_ode(eqn:x*'diff(y,x,2)-'diff(y,x,1)+a*y,y,x));
 656 [y = bessel_y(2,2*sqrt(a)*sqrt(x))*%k2*x
 657     +bessel_j(2,2*sqrt(a)*sqrt(x))*%k1*x];
 658 [method,ode_check(eqn,ans[1])];
 659 [odelin,0];
 660
 661 /* Kamke 2. 98  */
 662 assume(a>0);
 663 [a>0];
 664 (pn_(98),ans:contrib_ode(eqn:x*'diff(y,x,2)-'diff(y,x,1)-a*x^3*y,y,x));
 665 [y=%i*%k1*sinh(sqrt(a)*x^2/2)+%k2*cosh(sqrt(a)*x^2/2)];
 666 ode_check(eqn,ans[1]);
 667 0;
 668 method;
 669 xformtoconstcoeff;
 670 forget(a>0);
 671 [a>0];
 672
 673 /* Kamke 2. 99  */
 674 (pn_(99),ans:contrib_ode(eqn:x*'diff(y,x,2)-'diff(y,x,1)+x^3*(%e^x^3-v^2)*y,y,x));
 675 false;
 676
 677 /* Kamke 2. 100  */
 678 (pn_(100),ans:contrib_ode(eqn:x*'diff(y,x,2)+2*'diff(y,x,1)-x*y-%e^x,y,x));
 679 [y =((2*x-1)*%e^x)/(4*x)+(%k2*%e^x)/x+(%k1*%e^-x)/x];
 680 [method,ode_check(eqn,ans[1])];
 681 [kovacic,0];