share/contrib/diffequations/tests/rtestode_kamke_1_2.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Equations 101-200
   5
   6   Reference:
   7     E Kamke, Differentialgleichungen Losungsmethoden und Losungen,
   8     Vol 1, Geest & Portig, Leipzig, 1961
   9
  10     E. S. Cheb-Terrab, T. Koloknikov,  First Order ODEs,
  11     Symmetries and Linear Transformations, European Journal of
  12     Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003).
  13     (http://lie.uwaterloo.ca/papers/ode_iv.pdf)
  14 */
  15 /* Don't kill(all); It messes up trigsimp */
  16
  17 /* Print ode number*/
  18 (pn_(n_):=print("Kamke ODE 1.",n_),true);
  19 true;
  20
  21 /* 101 */
  22 (pn_(101),ans:contrib_ode(eqn:x*'diff(y,x)+x*y^2-y,y,x));
  23 [(x^2*y-2*x)/(2*y)=%c];
  24 solve(ans[1],y);
  25 [y=2*x/(x^2-2*%c)];
  26 [method,ode_check(eqn,%[1])];
  27 [exact,0];
  28
  29 /* 102 - FIXME: Why the complex solution? */
  30 assume(a>0);
  31 [a>0];
  32 (pn_(102),ans:contrib_ode(eqn:x*'diff(y,x)+x*y^2-y-a*x^3,y,x));
  33 [y=sqrt(a)*(%c*x*sinh(sqrt(a)*x^2/2)+%i*x*cosh(sqrt(a)*x^2/2))/(%i*sinh(sqrt(a)*x^2/2)+%c*cosh(sqrt(a)*x^2/2))];
  34 [method,ode_check(eqn,ans[1])];
  35 [riccati,0];
  36 forget(a>0);
  37 [a>0];
  38
  39 /* 103  - FIXME: Why the complex solution? */
  40 (pn_(103),ans:contrib_ode(eqn:x*'diff(y,x)+x*y^2-(2*x^2+1)*y-x^3,y,x));
  41 [y=((sqrt(2)*%c+%i)*x*sinh(sqrt(2)*x^2/2)+(%c+sqrt(2)*%i)*x*cosh(sqrt(2)*x^2/2))/(%i*sinh(sqrt(2)*x^2/2)+%c*cosh(sqrt(2)*x^2/2))];
  42 [method,ode_check(eqn,ans[1])];
  43 [riccati,0];
  44
  45 /* 104 */
  46 (pn_(104),ans:contrib_ode(eqn: x*'diff(y,x)+a*x*y^2+2*y+b*x,y,x));
  47 [y = (x*(%c*sqrt(a)*sqrt(-b)*%e^(2*sqrt(a)*sqrt(-b)*x)
  48                -sqrt(a)*sqrt(-b))
  49           -%c*%e^(2*sqrt(a)*sqrt(-b)*x)-1)
  50           /(a*x*(%c*%e^(2*sqrt(a)*sqrt(-b)*x)+1))];
  51 [method,ode_check(eqn,ans[1])];
  52 [riccati,0];
  53
  54 /* 105 */
  55 (pn_(105),ans:contrib_ode(eqn:x*'diff(y,x)+a*x*y^2+b*y+c*x+d,y,x));
  56 [y = (sqrt(a)*sqrt(-c)
  57                     *(%c*kummer_u(-((sqrt(a)*sqrt(-c)*d-b*c)/(2*c)),b,
  58                                   -2*sqrt(a)*sqrt(-c)*x)
  59                      -2*%c
  60                        *dkummer_u(-((sqrt(a)*sqrt(-c)*d-b*c)/(2*c)),b,
  61                                   -2*sqrt(a)*sqrt(-c)*x))
  62           +sqrt(a)*sqrt(-c)
  63                   *kummer_m(-((sqrt(a)*sqrt(-c)*d-b*c)/(2*c)),b,
  64                             -2*sqrt(a)*sqrt(-c)*x)
  65           -2*sqrt(a)*sqrt(-c)
  66             *dkummer_m(-((sqrt(a)*sqrt(-c)*d-b*c)/(2*c)),b,
  67                        -2*sqrt(a)*sqrt(-c)*x))
  68           /(a*(%c*kummer_u(-((sqrt(a)*sqrt(-c)*d-b*c)/(2*c)),b,
  69                            -2*sqrt(a)*sqrt(-c)*x)
  70               +kummer_m(-((sqrt(a)*sqrt(-c)*d-b*c)/(2*c)),b,
  71                         -2*sqrt(a)*sqrt(-c)*x)))];
  72 [method,ode_check(eqn,ans[1])];
  73 [riccati,0];
  74
  75 /* 106 */
  76 assume(not(equal(b+a,0)));
  77 [notequal(b+a,0)];
  78 (pn_("106a"),ans:contrib_ode(eqn:x*'diff(y,x)+x^a*y^2+(a-b)*y/2+x^b,y,x));
  79 [y=-((%c*x^(b/2+a/2)*sin(2*x^(b/2+a/2)/(b+a))-x^(b/2+a/2)*cos(2*x^(b/2+a/2)/(b+a)))/(x^a*sin(2*x^(b/2+a/2)/(b+a))+%c*x^a*cos(2*x^(b/2+a/2)/(b+a))))];
  80 [method,ode_check(eqn,ans[1])];
  81 [riccati,0];
  82 forget(notequal(b + a,0));
  83 [notequal(b+a,0)];
  84
  85 assume(equal(a+b,0),x>0);
  86 [equal(b + a, 0),x>0];
  87 (pn_("106b"),ans:contrib_ode(eqn:x*'diff(y,x)+x^a*y^2+(a-b)*y/2+x^b,y,x));
  88 [y = -((%c*sin(log(x))-cos(log(x)))/(x^a*sin(log(x))+%c*x^a*cos(log(x))))];
  89 [method,subst(-a,b,ode_check(eqn,ans[1]))];  /* Need to use a+b=0 */
  90 [riccati,0];
  91 forget(equal(a+b,0),x>0);
  92 [equal(b + a, 0),x>0];
  93
  94 /* 107 */
  95 (pn_(107),ans:contrib_ode(eqn:x*'diff(y,x)+a*x^alpha*y^2+b*y-c*x^beta,y,x));
  96 [[y = 'diff(%u,x,1)*x^(1-alpha)/(%u*a),%u*a^2*c*x^(beta+2*(alpha-1)-1)-'diff(%u,x,2)*a*x^(alpha-1)-'diff(%u,x,1)*(a*b*x^(alpha-2)-a*(alpha-1)*x^(alpha-2))=0]];
  97 method;
  98 riccati;
  99
 100 /* 108 */
 101 (pn_(108),ans:contrib_ode(eqn:x*'diff(y,x)-y^2*log(x)+y,y,x));
 102 [y=1/(x*(log(x)/x+1/x+%c))];
 103 [method,ode_check(eqn,ans[1])];
 104 [bernoulli,0];
 105
 106 /* 109 */
 107 (pn_(109),ans:contrib_ode(eqn:x*'diff(y,x)-y*(2*y*log(x)-1),y,x));
 108 [y=1/(x*(%c-2*(-log(x)/x-1/x)))];
 109 [method,ode_check(eqn,ans[1])];
 110 [bernoulli,0];
 111
 112 /* 110 Why the complex solution? */
 113 (pn_(110),ans:contrib_ode(eqn:x*'diff(y,x)+ f(x)*(y^2-x^2)-y,y,x));
 114 [y=(%c*x*sinh('integrate(f(x),x))+%i*x*cosh('integrate(f(x),x)))/(%i*sinh('integrate(f(x),x))+%c*cosh('integrate(f(x),x)))];
 115 [method,ode_check(eqn,ans[1])];
 116 [riccati,0];
 117
 118 /* 111 */
 119 (pn_(111),ans:contrib_ode(eqn: x*'diff(y,x) + y^3 + 3*x*y^2,y,x));
 120 false;
 121
 122 /* 112 */
 123 (pn_(112),ans:contrib_ode(eqn:x*'diff(y,x)-sqrt(y^2+x^2)-y,y,x));
 124 [x = %c*%e^(x*asinh(y/x)/abs(x))];
 125 assume(x>0);
 126 [x>0];
 127 solve(map(log,ans[1]/%c),y);
 128 [y = x*sinh(log(x/%c))];
 129 [method,ode_check(eqn,%[1])];
 130 [genhom,0];
 131 forget(x>0);
 132 [x>0];
 133
 134 /* 113 */
 135 (pn_(113),ans:contrib_ode(eqn:x*'diff(y,x)+a*sqrt(y^2+x^2)-y,y,x));
 136 [x=%c*%e^-(x*asinh(y/x)/(a*abs(x)))];
 137 assume(x>0);
 138 [x>0];
 139 solve(map(log,ans[1]/%c),y);
 140 [y=-x*sinh(a*log(x/%c))];
 141 [method,ode_check(eqn,%[1])];
 142 [genhom,0];
 143 forget(x>0);
 144 [x>0];
 145
 146 /* 114 */
 147 assume(x>0,y>0);
 148 [x>0,y>0];
 149 (pn_(114),ans:contrib_ode(eqn:x*'diff(y,x)-x*sqrt(y^2+x^2)-y,y,x));
 150 [x-asinh(y/x)=%c];
 151 solve(ans[1],y);
 152 [y=x*sinh(x-%c)];
 153 [method,ode_check(eqn,%[1])];
 154 [lie,0];
 155 forget(x>0,y>0);
 156 [x>0,y>0];
 157
 158 /* 115 - "solution" has changed - Jan 2007.  Don't care at present */
 159 /*
 160 (pn_(115),ans:contrib_ode(eqn:x*'diff(y,x)-x*(y-x)*sqrt(y^2+x^2)-y,y,x));
 161 [(2*'integrate((sqrt(y^2+x^2)*((y^2-x*y)*'integrate((2*y^2-x*y+x^2)*%e^-(log(y^2+x^2)/2)/(y^4-2*x*y^3+2*x^2*y^2-2*x^3*y+x^4),x)+(x-y)*'integrate(%e^-(log(y^2+x^2)/2)/(y-x),x))-x)/((y-x)*sqrt(y^2+x^2)),y)+2*y*'integrate(%e^-(log(y^2+x^2)/2)/(y-x),x)+x^2)/2=%c];
 162 method;
 163 lie;
 164 */
 165
 166 /* 116 - "solution" has changed - Jan 2007.  Don't care at present */
 167 /*
 168 (pn_(116),ans:contrib_ode(eqn:x*'diff(y,x)-x*sqrt((y^2-x^2)*(y^2-4*x^2))-y,y,x));
 169 [(2*'integrate((sqrt(y^4-5*x^2*y^2+4*x^4)*(y*'integrate((2*y^3-5*x^2*y)*%e^(-log(y+2*x)/2-log(y+x)/2-log(y-x)/2-log(y-2*x)/2)/(y^4-5*x^2*y^2+4*x^4),x)-'integrate(%e^(-log(y+2*x)/2-log(y+x)/2-log(y-x)/2-log(y-2*x)/2),x))-x)/sqrt(y^4-5*x^2*y^2+4*x^4),y)+2*y*'integrate(%e^(-log(y+2*x)/2-log(y+x)/2-log(y-x)/2-log(y-2*x)/2),x)+x^2)/2=%c];
 170 method;
 171 lie;
 172 */
 173
 174 /* 117 */
 175 (pn_(117),ans:contrib_ode(eqn:x*'diff(y,x)-x*exp(y/x)-y-x,y,x));
 176 [%c*x=%e^-((x*log(%e^(y/x)+1)-y)/x)];
 177 method;
 178 homogeneous;
 179
 180 /* 118 */
 181 (pn_(118),ans:contrib_ode(eqn:x*'diff(y,x)-y*log(y),y,x));
 182 [log(log(y))=log(x)+%c];
 183 solve(ans[1],y);
 184 [y=%e^(%e^%c*x)];
 185 [method,ode_check(eqn,%[1])];
 186 [separable,0];
 187
 188 /* 119 */
 189 (pn_(119),ans:contrib_ode(eqn:x*'diff(y,x)-y*(log(x*y)-1),y,x));
 190 [x = %c*log(x*y)];
 191 solve(ans[1],y);
 192 [y=%e^(x/%c)/x];
 193 [method,ode_check(eqn,%[1])];
 194 [genhom,0];
 195
 196 /* 120  Error return from ode1_lie - symmetries exist, see rtest_ode_sym.mac */
 197 /* (pn_(120),ans:contrib_ode(eqn:x*'diff(y,x)-y*(x*log(x^2/y)+2),y,x)); */
 198
 199 /* 121 */
 200 (pn_(121),ans:contrib_ode(eqn:x*'diff(y,x)+sin(y-x),y,x));
 201 false;
 202
 203 /* 122 */
 204 (pn_(122),ans:contrib_ode(eqn:x*'diff(y,x)+(sin(y)-3*x^2*cos(y))*cos(y),y,x));
 205 false;
 206
 207 /* 123 */
 208 (pn_(123),ans:contrib_ode(eqn:x*'diff(y,x)-x*sin(y/x)-y,y,x));
 209 [%c*x=%e^-((log(cos(y/x)+1)-log(cos(y/x)-1))/2)];
 210 method;
 211 homogeneous;
 212
 213 /* 124 */
 214 (pn_(124),ans:contrib_ode(eqn:x*'diff(y,x)+x*cos(y/x)-y+x,y,x));
 215 [%c*x=%e^-(sin(y/x)/(cos(y/x)+1))];
 216 method;
 217 homogeneous;
 218
 219 /* 125 */
 220 (pn_(125),ans:contrib_ode(eqn:x*'diff(y,x)+x*tan(y/x)-y,y,x));
 221 [%c*x=1/sin(y/x)];
 222 solve(ans[1],y);
 223 [y = asin(1/(%c*x))*x];
 224 [method,ode_check(eqn,%[1])];
 225 [homogeneous,0];
 226
 227 /* Equations 126 to 128 need pdiff */
 228 use_pdiff:true;
 229 true;
 230
 231 /* 126 */
 232 (pn_(126),ans:contrib_ode(eqn:x*'diff(y,x)-y*f(x*y),y,x));
 233 [y*'integrate(f(x*y)/(x*y*f(x*y)+x*y),x)-'integrate(((y^2*f(x*y)+y^2)*'integrate((x*y*pderivop(f,1)(x*y)-f(x*y)^2-f(x*y))/(x*y^2*f(x*y)^2+2*x*y^2*f(x*y)+x*y^2),x)+(y*f(x*y)+y)*'integrate(f(x*y)/(x*y*f(x*y)+x*y),x)+1)/(y*f(x*y)+y),y)=%c];
 234 method;
 235 lie;
 236
 237 /* 127  */
 238 (pn_(127),ans:contrib_ode(eqn:x*'diff(y,x)-y*f(x^a*y^b),y,x));
 239 [b*y*'integrate(f(x^a*y^b)/(b*x*y*f(x^a*y^b)+a*x*y),x)-'integrate(((b^2*y^2*f(x^a*y^b)+a*b*y^2)*'integrate((a*b*x^a*y^b*pderivop(f,1)(x^a*y^b)-b*f(x^a*y^b)^2-a*f(x^a*y^b))/(b^2*x*y^2*f(x^a*y^b)^2+2*a*b*x*y^2*f(x^a*y^b)+a^2*x*y^2),x)+(b^2*y*f(x^a*y^b)+a*b*y)*'integrate(f(x^a*y^b)/(b*x*y*f(x^a*y^b)+a*x*y),x)+b)/(b*y*f(x^a*y^b)+a*y),y)=%c];
 240 method;
 241 lie;
 242
 243 /* 128  */
 244 (pn_(128),ans:contrib_ode(eqn:x*'diff(y,x)+a*y-f(x)*g(x^a*y),y,x));
 245 ['integrate((x^a*f(x)*g(x^a*y)-a*x^a*y)/(x*g(x^a*y)),x)-'integrate((g(x^a*y)*'integrate((a*x^(2*a)*y*pderivop(g,1)(x^a*y)-a*x^a*g(x^a*y))/(x*g(x^a*y)^2),x)+x^a)/g(x^a*y),y)=%c];
 246 method;
 247 lie;
 248
 249 use_pdiff:false;
 250 false;
 251
 252 /* 129 */
 253 (pn_(129),ans:contrib_ode(eqn:(x+1)*'diff(y,x)+y*(y-x),y,x));
 254 [y = %e^x/((x+1)*(%c-%e^-1*expintegral_e(2,-x-1)/(x+1)))];
 255 [method,ode_check(eqn,ans[1])];
 256 [bernoulli,0];
 257
 258 /* 130 */
 259 (pn_(130),ans:contrib_ode(eqn:2*x*'diff(y,x)-y-2*x^3,y,x));
 260 [y=%e^(log(x)/2)*(2*%e^(5*log(x)/2)/5+%c)];
 261 [method,ode_check(eqn,ans[1])];
 262 [linear,0];
 263
 264 /*  131  */
 265 (pn_(131),ans:contrib_ode(eqn:(2*x+1)*'diff(y,x,1)-4*%e^-y+2,y,x));
 266 [-log(%e^y-2)/2=log(2*x+1)/2+%c];
 267 [method,ode_check(eqn,ans[1])];
 268 [separable,0];
 269
 270 /*  132  */
 271 (pn_(132),ans:contrib_ode(eqn:3*x*'diff(y,x,1)-3*x*log(x)*y^4-y,y,x));
 272 [-(((6*x^2*log(x)-3*x^2)*y^3+4*x)/(4*y^3))=%c];
 273 [method,ode_check(eqn,ans[1])];
 274 [exact,0];
 275
 276 /*  133  */
 277 (pn_(133),ans:contrib_ode(eqn:x^2*'diff(y,x,1)+y-x,y,x));
 278 [y = (%c + gamma_incomplete(0,1/x))*%e^(1/x)];
 279 [method,ode_check(eqn,ans[1])];
 280 [linear,0];
 281
 282 /*  134  */
 283 (pn_(134),ans:contrib_ode(eqn:x^2*'diff(y,x,1)-y+x^2*%e^(x-1/x),y,x));
 284 [y=%e^-(1/x)*(%c-%e^x)];
 285 [method,ode_check(eqn,ans[1])];
 286 [linear,0];
 287
 288 /*  135  */
 289 (pn_(135),ans:contrib_ode(eqn:x^2*'diff(y,x,1)-(x-1)*y,y,x));
 290 [y=%c*x*%e^(1/x)];
 291 [method,ode_check(eqn,ans[1])];
 292 [linear,0];
 293
 294 /*  136  */
 295 (pn_(136),ans:contrib_ode(eqn:x^2*'diff(y,x,1)+y^2+x*y+x^2,y,x));
 296 [%c*x=%e^(x/(y+x))];
 297 solve(ans[1],y);
 298 [y=-((x*log(%c*x)-x)/log(%c*x))];
 299 [method,ode_check(eqn,%[1])];
 300 [homogeneous,0];
 301
 302 /*  137  */
 303 (pn_(137),ans:contrib_ode(eqn:x^2*'diff(y,x,1)-y^2-x*y,y,x));
 304 [%c*x=%e^-(x/y)];
 305 solve(ans[1],y);
 306 [y=x/log(1/(%c*x))];
 307 [method,ode_check(eqn,%[1])];
 308 [homogeneous,0];
 309
 310 /*  138  */
 311 (pn_(138),ans:contrib_ode(eqn:x^2*'diff(y,x,1)-y^2-x*y-x^2,y,x));
 312 [%c*x=%e^atan(y/x)];
 313 solve(map(log,ans[1]),y);
 314 [y=x*tan(log(%c*x))];
 315 [method,ode_check(eqn,%[1])];
 316 [homogeneous,0];
 317
 318 /*  139  */
 319 (pn_(139),ans:contrib_ode(eqn:x^2*('diff(y,x,1)+y^2)+a*x^k-(b-1)*b,y,x));
 320 [[y='diff(%u,x,1)/%u,%u*(-a*x^(k-2)+b^2/x^2-b/x^2)-'diff(%u,x,2)=0]];
 321 method;
 322 riccati;
 323
 324 /*  140  */
 325 (pn_(140),ans:contrib_ode(eqn:x^2*('diff(y,x,1)+y^2)+4*x*y+2,y,x));
 326 [x=%c*(x*y+2)/(x*y+1)];
 327 solve(ans[1],y);
 328 [y=-((x-2*%c)/(x^2-%c*x))];
 329 [method,ode_check(eqn,%[1])];
 330 [genhom,0];
 331
 332 /*  141  */
 333 assume(4*b-a^2+2*a-1>0);
 334 [4*b-a^2+2*a-1>0];
 335 (pn_(141),ans:contrib_ode(eqn:x^2*('diff(y,x,1)+y^2)+a*x*y+b,y,x));
 336 [x=%c*%e^-(2*atan((2*x*y+a-1)/sqrt(4*b-a^2+2*a-1))/sqrt(4*b-a^2+2*a-1))];
 337 map(log,ans[1]/%c);
 338 log(x/%c)=-2*atan((2*x*y+a-1)/sqrt(4*b-a^2+2*a-1))/sqrt(4*b-a^2+2*a-1);
 339 solve(%,y);
 340 [y=-((sqrt(4*b-a^2+2*a-1)*tan(sqrt(4*b-a^2+2*a-1)*log(x/%c)/2)+a-1)/(2*x))];
 341 [method,ode_check(eqn,%[1])];
 342 [genhom,0];
 343 forget(4*b-a^2+2*a-1>0);
 344 [4*b-a^2+2*a-1>0];
 345
 346 /*  142  */
 347 (pn_(142),ans:contrib_ode(eqn:x^2*('diff(y,x,1)-y^2)-a*x^2*y+a*x+2,y,x));
 348 [y = -((%c*a^3*x^3*%e^(a*x)-%c*a^2*x^2*%e^(a*x)+2*%c*a*x*%e^(a*x)
 349                                  -2*%c*%e^(a*x)-1)
 350           /(x*(2*%c*%e^(a*x)+1)+%c*a^2*x^3*%e^(a*x)-2*%c*a*x^2*%e^(a*x)))];
 351 [method,ode_check(eqn,ans[1])];
 352 [riccati,0];
 353
 354
 355 /*  143  */
 356 assume(4*a*b>-1);
 357 [a*b>-(1/4)];
 358 (pn_(143),ans:contrib_ode(eqn:x^2*('diff(y,x,1)+a*y^2)-b,y,x));
 359 [x=%c*%e^-((log(-((-2*a*x*y+sqrt(4*a*b+1)+1)/(2*a*x*y+sqrt(4*a*b+1)-1)))
 360   /sqrt(4*a*b+1)))];
 361 method;
 362 genhom;
 363 forget(4*a*b>-1);
 364 [a*b>-(1/4)];
 365
 366 /*  144  */
 367 (pn_(144),ans:contrib_ode(eqn:x^2*('diff(y,x,1)+a*y^2)+b*x^alpha+c,y,x));
 368 [[y='diff(%u,x,1)/(%u*a),%u*a^2*(-b*x^(alpha-2)-c/x^2)-'diff(%u,x,2)*a=0]];
 369 method;
 370 riccati;
 371
 372 /*  145  */
 373 (pn_(145),ans:contrib_ode(eqn:x^2*'diff(y,x,1)+a*y^3-a*x^2*y^2,y,x));
 374 false;
 375
 376 /*  146  */
 377 (pn_(146),ans:contrib_ode(eqn:x^2*'diff(y,x,1)+x*y^3+a*y^2,y,x));
 378 false;
 379
 380 /*  147  */
 381 (pn_(147),ans:contrib_ode(eqn:x^2*'diff(y,x,1)+a*x^2*y^3+b*y^2,y,x));
 382 false;
 383
 384 /*  148  */
 385 (pn_(148),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,1)+x*y-1,y,x));
 386 [y=(asinh(x)+%c)*%e^-(log(x^2+1)/2)];
 387 [method,radcan(ode_check(eqn,ans[1]))];
 388 [linear,0];
 389
 390 /*  149  */
 391 (pn_(149),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,1)+x*y-x*(x^2+1),y,x));
 392 [y=((x^2+1)^(3/2)/3+%c)*%e^-(log(x^2+1)/2)];
 393 [method,radcan(ode_check(eqn,ans[1]))];
 394 [linear,0];
 395
 396 /*  150  */
 397 (pn_(150),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,1)+2*x*y-2*x^2,y,x));
 398 [y=(2*x^3/3+%c)/(x^2+1)];
 399 [method,ode_check(eqn,ans[1])];
 400 [linear,0];
 401
 402 /*  151  */
 403 (pn_(151),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,1)+(2*x*y-1)*(y^2+1),y,x));
 404 false;
 405
 406 /*  152  */
 407 (pn_(152),ans:contrib_ode(eqn:(x^2+1)*'diff(y,x,1)+x*cos(y)*sin(y)-x*(x^2+1)*cos(y)^2,y,x));
 408 false;
 409
 410 /*  153  */
 411 (pn_(153),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,1)-x*y+a,y,x));
 412 [y=(a*x/sqrt(x^2-1)+%c)*%e^(log(x^2-1)/2)];
 413 [method,radcan(ode_check(eqn,ans[1]))];
 414 [linear,0];
 415
 416 /*  154  */
 417 (pn_(154),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,1)+2*x*y-cos(x),y,x));
 418 [y=(sin(x)+%c)/(x^2-1)];
 419 [method,ode_check(eqn,ans[1])];
 420 [linear,0];
 421
 422 /*  155  */
 423 (pn_(155),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,1)+y^2-2*x*y+1,y,x),0);
 424 0$
 425 /*[y = (x^2*((x-1)*(%c*log(x+1)-%c*log(x-1)+2*%c+1)
 426                  +(x+1)*(%c*log(x+1)-%c*log(x-1)-2*%c+1))
 427           +(x+1)*(-%c*log(x+1)+%c*log(x-1)+2*%c-1)
 428           +(x-1)*(-%c*log(x+1)+%c*log(x-1)-2*%c-1))
 429           /((x-1)*(x+1)*(2*%c*log(x+1)-2*%c*log(x-1)+2))]; */
 430 [method,ode_check(eqn,ans[1])];
 431 [riccati,0];
 432
 433 /*  156  */
 434 (pn_(156),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,1)-y*(y-x),y,x));
 435 [y=%e^-(log(x^2-1)/2)/(x/sqrt(x^2-1)+%c)];
 436 [method,radcan(ode_check(eqn,ans[1]))];
 437 [bernoulli,0];
 438
 439 /*  157  */
 440 (pn_(157),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,1)+a*(y^2-2*x*y+1),y,x));
 441 [y = -(((%c*dgauss_b(1-a,a,1,-((x-1)/2))+dgauss_a(1-a,a,1,-((x-1)/2)))*x^2
 442        +(-2*%c*gauss_b(1-a,a,1,-((x-1)/2))-2*gauss_a(1-a,a,1,-((x-1)/2)))*a*x
 443        -%c*dgauss_b(1-a,a,1,-((x-1)/2))-dgauss_a(1-a,a,1,-((x-1)/2)))
 444        /((2*%c*gauss_b(1-a,a,1,-((x-1)/2))+2*gauss_a(1-a,a,1,-((x-1)/2)))*a))];
 445 [method,ode_check(eqn,ans[1])];
 446 [riccati,0];
 447
 448 /*  158  */
 449 (pn_(158),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,1)+a*x*y^2+x*y,y,x));
 450 [log(a*y+1)-log(y)=(log(x+1)+log(x-1))/2+%c];
 451 [method,ode_check(eqn,ans[1])];
 452 [separable,0];
 453
 454 /*  159  */
 455 (pn_(159),ans:contrib_ode(eqn:(x^2-1)*'diff(y,x,1)-2*x*y*log(y),y,x));
 456 [log(log(y))/2=(log(x+1)+log(x-1))/2+%c];
 457 [method,ode_check(eqn,ans[1])];
 458 [separable,0];
 459
 460 /*  160  */
 461 (pn_(160),ans:contrib_ode(eqn:(x^2-4)*'diff(y,x,1)+(x+2)*y^2-4*y,y,x));
 462 [y=%e^(4*(log(x-2)/4-log(x+2)/4))/(3*log(x+2)/4-4*(-log(x+2)/16+log(x-2)/16+1/(4*x+8))+log(x-2)/4+1/(x+2)+%c)];
 463 [method,ode_check(eqn,ans[1])];
 464 [bernoulli,0];
 465
 466 /*  161  */
 467 (pn_(161),ans:contrib_ode(eqn:(x^2-5*x+6)*'diff(y,x,1)+3*x*y-8*y+x^2,y,x));
 468 [y=(%c-(3*x^4-8*x^3)/12)/((x-3)*(x-2)^2)];
 469 [method,ode_check(eqn,ans[1])];
 470 [linear,0];
 471
 472 /*  162  */
 473 (pn_(162),ans:contrib_ode(eqn:(x-a)*(x-b)*'diff(y,x,1)+y^2+k*(y+x-a)*(y+x-b),y,x));
 474 [y = -((k*(-b*(x-b)^k-%c*a*(x-a)^k)+k*x*((x-b)^k+%c*(x-a)^k))
 475           /(k*((x-b)^k+%c*(x-a)^k)+(x-b)^k+%c*(x-a)^k))];
 476 [method,ode_check(eqn,ans[1])];
 477 [riccati,0];
 478
 479 /*  163  */
 480 assume(x>0);
 481 [x>0];
 482 (pn_(163),ans:contrib_ode(eqn:2*x^2*'diff(y,x,1)-2*y^2-x*y+2*a^2*x,y,x));
 483 [y=-((%c*a*sinh(2*a/sqrt(x))-%i*a*cosh(2*a/sqrt(x)))*sqrt(x)/(%i*sinh(2*a/sqrt(x))-%c*cosh(2*a/sqrt(x))))];
 484 [method,ode_check(eqn,ans[1])];
 485 [riccati,0];
 486 forget(x>0);
 487 [x>0];
 488
 489 /*  164  */
 490 (pn_(164),ans:contrib_ode(eqn:2*x^2*'diff(y,x,1)-2*y^2-3*x*y+2*a^2*x,y,x),0);
 491 0$
 492 /*
 493 [y=-(((bessel_y(-(1/2),2*a/sqrt(-x))*%c+bessel_j(-(1/2),2*a/sqrt(-x)))*x
 494    +((2*bessel_y(-(3/2),2*a/sqrt(-x))-2*bessel_y(1/2,2*a/sqrt(-x)))*%c
 495     -2*bessel_j(1/2,2*a/sqrt(-x))+2*bessel_j(-(3/2),2*a/sqrt(-x)))*a*sqrt(-x))
 496     /(4*bessel_y(-(1/2),2*a/sqrt(-x))*%c+4*bessel_j(-(1/2),2*a/sqrt(-x))))];
 497  */
 498 /* FIXME 2015-04-06  Form of solution has changed.
 499                      ode_check() crashes with stack overflow
 500 [method,ode_check(eqn,ans[1])];
 501 [riccati,0];
 502 */
 503
 504 /*  165  */
 505 (pn_(165),ans:contrib_ode(eqn:x*(2*x-1)*'diff(y,x,1)+y^2+(-4*x-1)*y+4*x,y,x));
 506 [(x*y-2*x^2)/(y-1)=%c];
 507 [method,ode_check(eqn,ans[1])];
 508 [exact,0];
 509
 510 /*  166  */
 511 (pn_(166),ans:contrib_ode(eqn:2*(x-1)*x*'diff(y,x,1)+(x-1)*y^2-x,y,x));
 512 [y = ((2*dgauss_b(1/2,3/2,1,x)*%c+2*dgauss_a(1/2,3/2,1,x))*x^2
 513           +(2*gauss_b(1/2,3/2,1,x)*%c-2*dgauss_b(1/2,3/2,1,x)*%c
 514                                      +2*gauss_a(1/2,3/2,1,x)
 515                                      -2*dgauss_a(1/2,3/2,1,x))
 516            *x)
 517           /((gauss_b(1/2,3/2,1,x)*%c+gauss_a(1/2,3/2,1,x))*x
 518            -gauss_b(1/2,3/2,1,x)*%c-gauss_a(1/2,3/2,1,x))];
 519 [method,ode_check(eqn,ans[1])];
 520 [riccati,0];
 521
 522 /*  167  */
 523 (pn_(167),ans:contrib_ode(eqn:3*x^2*'diff(y,x,1)-7*y^2-3*x*y-x^2,y,x));
 524 [%c*x=%e^(3*atan(7*y/(sqrt(7)*x))/sqrt(7))];
 525 solve(map(log,ans[1]),y);
 526 [y=sqrt(7)*x*tan(sqrt(7)*log(%c*x)/3)/7];
 527 [method,ode_check(eqn,%[1])];
 528 [homogeneous,0];
 529
 530 /*  168  */
 531 (pn_(168),ans:contrib_ode(eqn:3*(x^2-4)*'diff(y,x,1)+y^2-x*y-3,y,x),0);
 532 0$
 533 /*
 534 [y = -( ((3*dgauss_b(1/2,7/6,4/3,-((x-2)/4))*%c
 535          +3*dgauss_a(1/2,7/6,4/3,-((x-2)/4)))
 536           *x^2
 537      +(-6*gauss_b(1/2,7/6,4/3,-((x-2)/4))*%c
 538        -6*gauss_a(1/2,7/6,4/3,-((x-2)/4)))
 539            *x-12*dgauss_b(1/2,7/6,4/3,-((x-2)/4))*%c
 540           -12*dgauss_a(1/2,7/6,4/3,-((x-2)/4)))
 541    /(4*gauss_b(1/2,7/6,4/3,-((x-2)/4))*%c
 542     +4*gauss_a(1/2,7/6,4/3,-((x-2)/4))) )  ];
 543  */
 544 [method,ode_check(eqn,ans[1])];
 545 [riccati,0];
 546
 547 /*  169  */
 548 (pn_(169),ans:contrib_ode(eqn:(a*x+b)^2*'diff(y,x,1)+(a*x+b)*y^3+c*y^2,y,x));
 549 false;
 550
 551 /*  170  */
 552 (pn_(170),ans:contrib_ode(eqn:x^3*'diff(y,x,1)-y^2-x^4,y,x));
 553 [x=%c*%e^-(x^2/(y-x^2))];
 554  solve(ans[1],y);
 555 [y=(log(%c/x)+1)*x^2/log(%c/x)];
 556 [method,ode_check(eqn,%[1])];
 557 [genhom,0];
 558
 559 /*  171  */
 560 (pn_(171),ans:contrib_ode(eqn:x^3*'diff(y,x,1)-y^2-x^2*y,y,x));
 561 [y=x/(1/x+%c)];
 562 [method,ode_check(eqn,ans[1])];
 563 [bernoulli,0];
 564
 565 /*  172  */
 566 (pn_(172),ans:contrib_ode(eqn:x^3*'diff(y,x,1)-x^4*y^2+x^2*y+20,y,x));
 567 [x=%c*%e^-((log(x^2*y+5)-log(x^2*y-4))/9)];
 568 method;
 569 genhom;
 570
 571 /*  173  */
 572 (pn_(173),ans:contrib_ode(eqn:x^3*'diff(y,x,1)-x^6*y^2+(3-2*x)*x^2*y+3,y,x));
 573 [y=-((3*%e^(4*x)-%c)/(x^3*%e^(4*x)+%c*x^3))];
 574 [method,ode_check(eqn,ans[1])];
 575 [riccati,0];
 576
 577 /*  174  */
 578 (pn_(174),ans:contrib_ode(eqn:x*(x^2+1)*'diff(y,x,1)+x^2*y,y,x));
 579 [y=%c*%e^-(log(x^2+1)/2)];
 580 [method,ode_check(eqn,ans[1])];
 581 [linear,0];
 582
 583 /*  175  */
 584 (pn_(175),ans:contrib_ode(eqn:x*(x^2-1)*'diff(y,x,1)-(2*x^2-1)*y+a*x^3,y,x));
 585 [y=x*(a/sqrt(x^2-1)+%c)*%e^(log(x+1)/2+log(x-1)/2)];
 586 [method,radcan(ode_check(eqn,ans[1]))];
 587 [linear,0];
 588
 589 /*  176  */
 590 (pn_(176),ans:contrib_ode(eqn:x*(x^2-1)*'diff(y,x,1)+(x^2-1)*y^2-x^2,y,x));
 591 [[y='diff(%u,x,1)*x/%u,%u/(x*(x^2-1))-'diff(%u,x,2)/x-'diff(%u,x,1)/x^2=0]];
 592 method;
 593 riccati;
 594
 595 /*  177  */
 596 (pn_(177),ans:contrib_ode(eqn:(x-1)*x^2*'diff(y,x,1)-y^2-(x-2)*x*y,y,x));
 597 [y=(x-1)*%e^-(2*(log(x-1)-log(x)))/((2*x-1)/(2*x^2-4*x+2)-1/(2*x^2-4*x+2)+%c)];
 598 [method,ode_check(eqn,ans[1])];
 599 [bernoulli,0];
 600
 601 /*  178  */
 602 (pn_(178),ans:contrib_ode(eqn:2*x*(x^2-1)*'diff(y,x,1)+2*(x^2-1)*y^2+(5-3*x^2)*y+x^2-3,y,x));
 603 [y = (%c*sqrt(x-1)*sqrt(x+1)*'integrate(1/(sqrt(x-1)*sqrt(x)*sqrt(x+1)),x)
 604          +sqrt(x-1)*sqrt(x+1)+%c*sqrt(x))
 605     /(%c*sqrt(x-1)*sqrt(x+1)*'integrate(1/(sqrt(x-1)*sqrt(x)*sqrt(x+1)),x)
 606          +sqrt(x-1)*sqrt(x+1))];
 607 [method,ode_check(eqn,ans[1])];
 608 [riccati,0];
 609
 610 /*  179  */
 611 (pn_(179),ans:contrib_ode(eqn:3*x*(x^2-1)*'diff(y,x,1)+x*y^2+(-x^2-1)*y-3*x,y,x));
 612 [[y='diff(%u,x,1)*(3*x^2-3)/%u,-'diff(%u,x,1)*(6*x/(3*x^2-3)^2-(x^2/(3*x^3-3*x)+1/(3*x^3-3*x))/(3*x^2-3))-'diff(%u,x,2)/(3*x^2-3)+%u/((x^2-1)*(3*x^2-3)^2)=0]];
 613 method;
 614 riccati;
 615
 616 /*  180  */
 617 assume(4*a*c>b^2);
 618 [4*a*c>b^2];
 619 (pn_(180),ans:contrib_ode(eqn:(a*x^2+b*x+c)*(x*'diff(y,x,1)-y)-y^2+x^2,y,x));
 620 [y=-((%c*x*sinh(2*atan((2*a*x+b)/sqrt(4*a*c-b^2))/sqrt(4*a*c-b^2))+%i*x*cosh(2*atan((2*a*x+b)/sqrt(4*a*c-b^2))/sqrt(4*a*c-b^2)))/(%i*sinh(2*atan((2*a*x+b)/sqrt(4*a*c-b^2))/sqrt(4*a*c-b^2))+%c*cosh(2*atan((2*a*x+b)/sqrt(4*a*c-b^2))/sqrt(4*a*c-b^2))))];
 621 [method,ode_check(eqn,ans[1])];
 622 [riccati,0];
 623 forget(4*a*c>b^2);
 624 [4*a*c>b^2];
 625
 626 /*  181  */
 627 assume(a>0);
 628 [a>0];
 629 (pn_(181),ans:contrib_ode(eqn:x^4*('diff(y,x,1)+y^2)+a,y,x));
 630 [y=-((sqrt(a)-x*tan((sqrt(a)-%c*x)/x))/(x^2*tan((sqrt(a)-%c*x)/x)))];
 631 [method,ode_check(eqn,ans[1])];
 632 [riccati,0];
 633 forget(a>0);
 634 [a>0];
 635
 636 /*  182  */
 637 (pn_(182),ans:contrib_ode(eqn:x*(x^3-1)*'diff(y,x,1)-2*x*y^2+y+x^2,y,x));
 638  [y = (2*%c*(x-1)*x^6+x*(3*%c*(x-1)*(x^2+x+1)-2*(x-1))
 639                           +x^4*(2*(x-1)-3*%c*(x-1)*(x^2+x+1))
 640                           +x^5*(%c*(x^2+x+1)+%c*(x-1))
 641                           +x^2*(-%c*(x^2+x+1)-%c*(x-1))
 642                           +x^3*(x^2+x+(1-2*%c)*(x-1)+1)-x^2-2*x)
 643          /(3*%c*(x-1)*x^2*(x^2+x+1)+3*(x-1)*(x^2+x+1))];
 644 [method,ode_check(eqn,ans[1])];
 645 [riccati,0];
 646
 647 /*  183  */
 648 (pn_(183),ans:contrib_ode(eqn:(2*x^4-x)*'diff(y,x,1)-2*(x^3-1)*y,y,x));
 649 [y=%c*x^2*%e^-(log(2*x^3-1)/3)];
 650 [method,ode_check(eqn,ans[1])];
 651 [linear,0];
 652
 653 /*  184  - FIXME 2006-12-11 - Runs forever */
 654 /* (pn_(184),ans:contrib_ode(eqn:(a*x^2+b*x+c)^2*('diff(y,x,1)+y^2)+A,y,x));
 655 [[y='diff(%u,x,1)/%u,-%u*A/(a^2*x^4+2*a*b*x^3+2*a*c*x^2+b^2*x^2+2*b*c*x+c^2)-'diff(%u,x,2)=0]];
 656 method;
 657 riccati; */
 658
 659 /*  185  */
 660 (pn_( 185 ),ans:contrib_ode(eqn: x^7*'diff(y,x,1)+2*(x^2+1)*y^3+5*x^3*y^2 ,y,x));
 661 false;
 662
 663 /*  186  */
 664 (pn_(186),ans:contrib_ode(eqn:x^n*'diff(y,x,1)+y^2+(1-n)*x^(n-1)*y+x^(2*n-2),y,x));
 665 [x=%c*%e^-atan(x^(1-n)*y)];
 666 method;
 667 genhom;
 668
 669 /*  187  */
 670 assume(n^2-2*n-4*a*b+1>0);
 671 [n^2-2*n-4*a*b+1>0];
 672 (pn_(187),ans:contrib_ode(eqn:x^n*'diff(y,x,1)-a*y^2-b*x^(2*n-2),y,x));
 673 [x=%c*%e^(log(-((-2*a*x*y+sqrt(n^2-2*n-4*a*b+1)*x^n+(n-1)*x^n)/(2*a*x*y+sqrt(n^2-2*n-4*a*b+1)*x^n+(1-n)*x^n)))/sqrt(n^2-2*n-4*a*b+1))];
 674 method;
 675 genhom;
 676 forget(n^2-2*n-4*a*b+1>0);
 677 [n^2-2*n-4*a*b+1>0];
 678
 679 /* 188  Abel eqn
 680   Some choices that are integrable include
 681     (n:3, b:1, a:n+b);   => K = -27/4
 682     (n:7, b:2, a:n+b);   => K = -343/36
 683 */
 684 (pn_(188),n:3,b:1,a:n+b,ans:contrib_ode(eqn:x^(2*n+1)*'diff(y,x)-a*y^3-b*x^(3*n),y,x));
 685 [x=%c*%e^-(((2*y-x^3)*log((2*y-x^3)/x^3)+(x^3-2*y)*log((y+x^3)/x^3)+3*x^3)/(18*y-9*x^3))];
 686 method;
 687 genhom;
 688 kill(a,b,n);
 689 done;
 690
 691 /*  189  */
 692 declare(n,integer);
 693 done;
 694 (pn_(189),ans:contrib_ode(eqn:x^(n+m*(n-1))*'diff(y,x,1)-a*y^n-b*x^((m+1)*n),y,x),done);
 695 done;
 696 method;
 697 lie;
 698 remove(n,integer);
 699 done;
 700
 701 /*  190  */
 702 (pn_(190),ans:contrib_ode(eqn:sqrt(x^2-1)*'diff(y,x,1)-sqrt(y^2-1),y,x));
 703 [log(2*sqrt(y^2-1)+2*y)=log(2*sqrt(x^2-1)+2*x)+%c];
 704 [method,ode_check(eqn,ans[1])];
 705 [separable,0];
 706
 707 /*  191  */
 708 (pn_(191),ans:contrib_ode(eqn:sqrt(1-x^2)*'diff(y,x,1)-y*sqrt(y^2-1),y,x));
 709 [-asin(1/abs(y))=asin(x)+%c];
 710 [method,ode_check(eqn,ans[1])];
 711 [separable,0];
 712
 713 /*  192  */
 714 (pn_(192),ans:contrib_ode(eqn:sqrt(x^2+a^2)*'diff(y,x,1)+y-sqrt(x^2+a^2)+x,y,x));
 715 [y = %e^-asinh(x/abs(a))*(((a^2*log(sqrt(x^2/a^2+1)+x/abs(a)))/2
 716                                +(x*sqrt(x^2+a^2))/2)
 717                                /a
 718                                -((x*sqrt(x^2+a^2))/2
 719                                 -(a^2*log(sqrt(x^2/a^2+1)+x/abs(a)))/2)
 720                                 /abs(a)+((a-abs(a))*x^2)/(2*a*abs(a))+%c)]$
 721 assume(a>0);  /* Just check when a>0 */
 722 [a>0];
 723 logarc( expand(ans[1],0,0));
 724 y = (((a^2*log(sqrt(x^2/a^2+1)+x/a))/2+(x*sqrt(x^2+a^2))/2)/a
 725          -((x*sqrt(x^2+a^2))/2-(a^2*log(sqrt(x^2/a^2+1)+x/a))/2)/a+%c)
 726          /(sqrt(x^2/a^2+1)+x/a)$
 727 [method,ode_check(eqn,%)];
 728 [linear,0];
 729 forget(a>0);
 730 [a>0];
 731
 732 /*  193  */
 733 (pn_(193),ans:contrib_ode(eqn:x*log(x)*'diff(y,x,1)+y-a*x*(log(x)+1),y,x));
 734 [y=(a*(x*log(x)-x)+a*x+%c)/log(x)];
 735 [method,ode_check(eqn,ans[1])];
 736 [linear,0];
 737
 738 /*  194  */
 739 (pn_(194),ans:contrib_ode(eqn:x*log(x)*'diff(y,x,1)-log(x)*y^2+(-2*log(x)^2-1)*y-log(x)^3,y,x));
 740 [y=-((%c*log(x)^3+(2*%c+2)*log(x))/(%c*log(x)^2+2))];
 741 [method,ode_check(eqn,ans[1])];
 742 [riccati,0];
 743
 744 /*  195  */
 745 (pn_(195),ans:contrib_ode(eqn:sin(x)*'diff(y,x,1)-sin(x)^2*y^2+(cos(x)-3*sin(x))*y+4,y,x));
 746 [y=-((4*%e^(5*x)-%c)/((%e^(5*x)+%c)*sin(x)))];
 747 [method,ode_check(eqn,ans[1])];
 748 [riccati,0];
 749
 750 /*  196  */
 751 /* FIXME:  This test now prompts - 2006-10-02
 752 (pn_(196),ans:contrib_ode(eqn:cos(x)*'diff(y,x,1)+y+cos(x)*(sin(x)+1),y,x));
 753 [y = %e^(log(sin(x)-1)/2-log(sin(x)+1)/2)
 754      *('integrate((-sin(x)-1)*%e^(log(sin(x)+1)/2-log(sin(x)-1)/2),x)+%c)];
 755 ans:radcan(ans[1]);
 756 y=-sqrt(sin(x)-1)*('integrate((sin(x)+1)^(3/2)/sqrt(sin(x)-1),x)-%c)
 757   /sqrt(sin(x)+1);
 758 [method,ode_check(eqn,ans)];
 759 [linear,0];
 760 */
 761 /*  197  */
 762 (pn_(197),ans:contrib_ode(eqn:cos(x)*'diff(y,x,1)-y^4-sin(x)*y,y,x));
 763 [y=1/(cos(x)*(%c-3*(tan(x)^3/3+tan(x)))^(1/3))];
 764 [method,ode_check(eqn,ans[1])];
 765 [bernoulli,0];
 766
 767 /*  198  FIXME: This now fails.  May be a maxima bug.  Jan 2007 */
 768 /*
 769 (pn_(198),ans:contrib_ode(eqn:cos(x)*sin(x)*'diff(y,x,1)-y-sin(x)^3,y,x));
 770 [y=sin(x)*(%c-sqrt(sin(x)^2-1))*%e^-(log(sin(x)^2-1)/2)];
 771 [method,ode_check(eqn,ans[1])];
 772 [linear,0];
 773 */
 774
 775 /*  199  - also Murphy 1.129 */
 776 (pn_(199),ans:contrib_ode(eqn:sin(2*y)+sin(2*x)*'diff(y,x,1),y,x));
 777 [(log(cos(2*y)+1)-log(cos(2*y)-1))/4=%c-(log(cos(2*x)+1)-log(cos(2*x)-1))/4];
 778 trigsimp(map(exp,4*ans[1]));
 779 (cos(2*y)+1)/(cos(2*y)-1) = (%e^(4*%c)*cos(2*x)-%e^(4*%c))/(cos(2*x)+1);
 780 subst(%c,exp(4*%c),%);
 781 (cos(2*y)+1)/(cos(2*y)-1)=(%c*cos(2*x)-%c)/(cos(2*x)+1);
 782 solve(%,y);
 783 [y = acos(%c*cos(2*x)/(%c*cos(2*x)-cos(2*x)-%c-1)
 784     +cos(2*x)/(%c*cos(2*x)-cos(2*x)-%c-1)-%c/(%c*cos(2*x)-cos(2*x)-%c-1)
 785     +1/(%c*cos(2*x)-cos(2*x)-%c-1))/2];
 786 [method,ode_check(eqn,%[1])];
 787 [separable,0];
 788
 789 /*  200  */
 790 (pn_(200),ans:contrib_ode(eqn:(a*sin(x)^2+b)*'diff(y,x,1)+a*sin(2*x)*y+A*x*(a*sin(x)^2+c),y,x));
 791 [y=((A*a*(2*x*sin(2*x)+cos(2*x)-2*x^2))/8-(A*c*x^2)/2+%c)/(a*sin(x)^2+b)];
 792 [method,ode_check(eqn,ans[1])];
 793 [linear,0];