share/contrib/diffequations/tests/rtestode_kamke_1_5.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Equations 1.401-1.500
   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 */
  16 /* Don't kill(all); It messes up trigsimp */
  17
  18 /* Print ode number*/
  19 (pn_(n_):=print("Kamke ODE 1.",n_),true);
  20 true;
  21
  22 /*  401  */
  23 (pn_(401),ans:contrib_ode(eqn:3*('diff(y,x,1))^2-2*x*'diff(y,x,1)+y,y,x));
  24 [4*y^3-x^2*y^2-18*%c*x*y+4*%c*x^3+27*%c^2=0];
  25 method;
  26 lagrange;
  27
  28 /*  402  Why the complex solution? */
  29 (pn_(402),ans:contrib_ode(eqn:3*('diff(y,x,1))^2+4*x*'diff(y,x,1)-y+x^2,y,x));
  30 [(2^(3/2)*%i*sqrt(3*y+x^2)+sqrt(2)*%i*x)/3 = %c]$
  31 [method,ode_check(eqn,ans[1])];
  32 [lie,0];
  33
  34 /*  403  */
  35 (pn_(403),ans:contrib_ode(eqn:a*('diff(y,x,1))^2+b*'diff(y,x,1)-y,y,x));
  36 [[x=log(%t)*b+2*%t*a+%c,y=%t*b+%t^2*a]];
  37 [method,ode_check(eqn,ans[1])];
  38 [lagrange,0];
  39
  40 /*  404  */
  41 /* This equation presents a problem - runs forever */
  42 /* (pn_( 404 ),ans:contrib_ode(eqn: a*('diff(y,x,1))^2+b*x^2*'diff(y,x,1)+c*x*y ,y,x)); */
  43
  44 /*  405  */
  45 (pn_(405),ans:contrib_ode(eqn:a*('diff(y,x,1))^2+y*'diff(y,x,1)-x,y,x));
  46 false;
  47
  48 /*  406  */
  49 (pn_(406),ans:contrib_ode(eqn:a*('diff(y,x,1))^2-y*'diff(y,x,1)-x,y,x));
  50 false;
  51
  52 /*  407  */
  53 (pn_(407),ans:contrib_ode(eqn:x*('diff(y,x,1))^2-y,y,x));
  54 [x^2*(y^2+(-2*x-2*%c)*y+x^2-2*%c*x+%c^2)=0];
  55 method;
  56 lagrange;
  57
  58 /*  408  */
  59 (pn_( 408 ),ans:contrib_ode(eqn: x*('diff(y,x,1))^2-2*y+x ,y,x));
  60 [[x=%c*%e^-(2*(log(%t-1)-1/(%t-1))),y=(%t^2+1)*x/2]];
  61 [method,ode_check(eqn,ans[1])];
  62 [lagrange,0];
  63
  64 /*  409  */
  65 (pn_( 409 ),ans:contrib_ode(eqn: x*('diff(y,x,1))^2-2*'diff(y,x,1)-y ,y,x));
  66 [[x=(2*(%t-log(%t))+%c)/(%t-1)^2,y = %t^2*x-2*%t]];
  67 [method,ode_check(eqn,ans[1])];
  68 [lagrange,0];
  69
  70 /*  410  */
  71 /* This equation presents a problem - runs forever */
  72 /* (pn_( 410 ),ans:contrib_ode(eqn: x*('diff(y,x,1))^2+4*'diff(y,x,1)-2*y ,y,x)); */
  73
  74 /*  411  */
  75 (pn_(411),ans:contrib_ode(eqn:x*('diff(y,x,1))^2+x*'diff(y,x,1)-y ,y,x));
  76 [[x=%c*%e^(1/%t)/%t^2,y=(%t^2+%t)*x]];
  77 [method,ode_check(eqn,ans[1])];
  78 [lagrange,0];
  79
  80 /*  412  */
  81 (pn_(412),ans:contrib_ode(eqn:x*('diff(y,x,1))^2+y*'diff(y,x,1)+a,y,x));
  82 [(4*log(sqrt(y^2-4*a*x)+2*y)+2*log(sqrt(y^2-4*a*x)-y))/3=%c];
  83 [method,ode_check(eqn,ans[1])];
  84 [lie,0];
  85
  86 /*  413  Not sure about this one - ode_check() chokes */
  87 (pn_(413),ans:contrib_ode(eqn:x*('diff(y,x,1))^2+y*'diff(y,x,1)-x^2,y,x),0);
  88 0$
  89 /* FIXME 2015-04 solution has changed
  90 [-((120*'integrate((sqrt(y^2+4*x^3)*(y*%e^(3*log(y^2+4*x^3)/2)+(-y^3-4*x^3*y)*%e^(log(y^2+4*x^3)/2)+32*x^3*y^2+128*x^6)+(-4*y^2-32*x^3)*%e^(3*log(y^2+4*x^3)/2)+(4*y^4+16*x^3*y^2)*%e^(log(y^2+4*x^3)/2))/(sqrt(y^2+4*x^3)*((4*y^2+16*x^3)*%e^(3*log(y^2+4*x^3)/2)+(-4*y^4-16*x^3*y^2)*%e^(log(y^2+4*x^3)/2)+64*x^3*y^3+256*x^6*y)+(16*y^3+64*x^3*y)*%e^(3*log(y^2+4*x^3)/2)+(-16*y^5-64*x^3*y^3)*%e^(log(y^2+4*x^3)/2)+256*x^3*y^4+1024*x^6*y^2),y)-64*log(%e^(log(y^2+4*x^3)/2)+4*y)+25*log(%e^(log(y^2+4*x^3)/2)+y)+9*log(%e^(log(y^2+4*x^3)/2)-y)-75*log(x))/120) = %c];
  91  */
  92 method;
  93 lie;
  94
  95 /*  414  "Solution" changed Jan 2007.  Don't worry at present. */
  96 /* (pn_(414),ans:contrib_ode(eqn:x*('diff(y,x,1))^2+y*'diff(y,x,1)+x^3,y,x),0);
  97 0$
  98 /*
  99 [-(5*'integrate((sqrt(y^2-4*x^4)*'integrate(((52*x^3*y^2-8*x^7)*%e^(log(y+2*x^2)/2+log(y-2*x^2)/2)+20*x^3*y^3-80*x^7*y)/((65*y^5-270*x^4*y^3+40*x^8*y)*%e^(log(y+2*x^2)/2+log(y-2*x^2)/2)+97*y^6-464*x^4*y^4+306*x^8*y^2-8*x^12),x)+5*y*'integrate(((52*x^3*y^2-8*x^7)*%e^(log(y+2*x^2)/2+log(y-2*x^2)/2)+20*x^3*y^3-80*x^7*y)/((65*y^5-270*x^4*y^3+40*x^8*y)*%e^(log(y+2*x^2)/2+log(y-2*x^2)/2)+97*y^6-464*x^4*y^4+306*x^8*y^2-8*x^12),x)-2)/(sqrt(y^2-4*x^4)+5*y),y)+5*'integrate((24*y^2+4*x^4)/(25*x*y*%e^(log(y+2*x^2)/2+log(y-2*x^2)/2)+65*x*y^2-10*x^5),x)-3*log(x))/5=%c];
 100  */
 101 method;
 102 lie; */
 103
 104 /*  415  */
 105 (pn_(415),ans:contrib_ode(eqn:x*('diff(y,x,1))^2+y*'diff(y,x,1)-y^4,y,x));
 106 [2*log(sqrt(4*x*y^2+1)-1)-2*log(y)=%c];
 107 [method,ode_check(eqn,ans[1])];
 108 [lie,0];
 109
 110 /*  416  */
 111 (pn_(416),ans:contrib_ode(eqn:x*('diff(y,x,1))^2+(y-3*x)*'diff(y,x,1)+y,y,x));
 112 [[x=%c*(%t+1)*%e^-(3*log(%t)/2),y=-((%t^2-3*%t)*x/(%t+1))]];
 113 [method,ode_check(eqn,ans[1])];
 114 [lagrange,0];
 115
 116 /*  417  */
 117 (pn_(417),ans:contrib_ode(eqn:x*('diff(y,x,1))^2-y*'diff(y,x,1)+a,y,x));
 118 [y=(%c^2*x+a)/%c,y=-2*sqrt(a*x),y=2*sqrt(a*x)];
 119 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3])];
 120 [clairaut,0,0,0];
 121
 122 /*  418  */
 123 (pn_(418),ans:contrib_ode(eqn:x*('diff(y,x,1))^2-y*'diff(y,x,1)+a*y,y,x));
 124 [[x=%c*(%t-a)*%e^-(%t/a),y=%t^2*x/(%t-a)]];
 125 [method,ode_check(eqn,ans[1])];
 126 [lagrange,0];
 127
 128 /*  419  */
 129 (pn_(419),ans:contrib_ode(eqn:x*('diff(y,x,1))^2+2*y*'diff(y,x,1)-x,y,x));
 130 [[x=%c*%t*%e^-((2*log(3*%t^2-1)/3)),y=-((%t^2-1)*x/(2*%t))]];
 131 [method,ode_check(eqn,ans[1])];
 132 [lagrange,0];
 133
 134 /*  420  */
 135 (pn_(420),ans:contrib_ode(eqn:x*('diff(y,x,1))^2-2*y*'diff(y,x,1)+a,y,x));
 136 [(2*log(2*sqrt(y^2-a*x)-y)+4*log(sqrt(y^2-a*x)+y))/3=%c];
 137 [method,ode_check(eqn,ans[1])];
 138 [lie,0];
 139
 140 /*  421  */
 141 (pn_(421),ans:contrib_ode(eqn:x*('diff(y,x,1))^2-2*y*'diff(y,x,1)-x,y,x));
 142 [x*(2*%c*y-x^2+%c^2)=0];
 143 solve(ans[1],y);
 144 [y=(x^2-%c^2)/(2*%c)];
 145 [method,ode_check(eqn,%[1])];
 146 [lagrange,0];
 147
 148 /*  422  */
 149 (pn_(422),ans:contrib_ode(eqn:x*('diff(y,x,1))^2-2*y*'diff(y,x,1)+4*x,y,x));
 150 [x*(2*%c*y-x^2-4*%c^2)=0];
 151 solve(ans[1],y);
 152 [y=(x^2+4*%c^2)/(2*%c)];
 153 [method,ode_check(eqn,%[1])];
 154 [lagrange,0];
 155
 156
 157 /*  423  */
 158 (pn_(423),ans:contrib_ode(eqn:x*('diff(y,x,1))^2-2*y*'diff(y,x,1)+2*y+x,y,x));
 159 [x*(2*%c*y-x^2-2*%c*x-2*%c^2)=0];
 160 solve(ans[1],y);
 161 [y=(x^2+2*%c*x+2*%c^2)/(2*%c)];
 162 [method,ode_check(eqn,%[1])];
 163 [lagrange,0];
 164
 165 /*  424  */
 166 (pn_(424),ans:contrib_ode(eqn:x*('diff(y,x,1))^2+a*y*'diff(y,x,1)+b*x,y,x));
 167 [[x=%c*%t*%e^-((a+2)*log(b+%t^2*(a+1))/(2*a+2)),y=-((b+%t^2)*x/(%t*a))]];
 168 [method,ode_check(eqn,%[1])];
 169 [lagrange,0];
 170
 171 /*  425  */
 172 (pn_(425),ans:contrib_ode(eqn: (x+1)*('diff(y,x,1))^2-(y+x)*'diff(y,x,1)+y ,y,x));
 173 [y=((%c^2-%c)*x+%c^2)/(%c-1),y=-2*sqrt(x+1)+x+2,y=2*sqrt(x+1)+x+2];
 174 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3])];
 175 [clairaut,0,0,0];
 176
 177 /*  426  */
 178 (pn_(426),ans:contrib_ode(eqn:(3*x+1)*('diff(y,x,1))^2-3*(y+2)*'diff(y,x,1)+9 ,y,x));
 179 [y=(3*%c^2*x+%c^2-6*%c+9)/(3*%c),y=-2*sqrt(3*x+1)-2,y=2*sqrt(3*x+1)-2];
 180 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3])];
 181 [clairaut,0,0,0];
 182
 183 /*  427  */
 184 (pn_(427),ans:contrib_ode(eqn: (3*x+5)*('diff(y,x,1))^2-(3*y+x)*'diff(y,x,1)+y ,y,x));
 185 [y=((3*%c^2-%c)*x+5*%c^2)/(3*%c-1),y=-((2*sqrt(5)*sqrt(3*x+5)-3*x-10)/9),y=(2*sqrt(5)*sqrt(3*x+5)+3*x+10)/9];
 186 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3])];
 187 [clairaut,0,0,0];
 188
 189 /*  428  */
 190 /* This equation presents a problem - runs forever */
 191 (pn_(428),ans:contrib_ode(eqn:a*x*('diff(y,x,1))^2+(-a*y+b*x+c)*'diff(y,x,1)-b*y ,y,x));
 192 [y=((%c*b+%c^2*a)*x+%c*c)/(b+%c*a),y=-((2*sqrt(-b*c*x)+b*x-c)/a),y=(2*sqrt(-b*c*x)-b*x+c)/a];
 193 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3])];
 194 [clairaut,0,0,0];
 195
 196 /*  429  */
 197 (pn_(429),ans:contrib_ode(eqn: a*x*('diff(y,x,1))^2-(a*y+b*x-b-a)*'diff(y,x,1)+b*y ,y,x));
 198 [y=((%c*b-%c^2*a)*x-%c*b-%c*a)/(b-%c*a),y=-((2*sqrt((b^2+a*b)*x)-b*x-b-a)/a),y=(2*sqrt((b^2+a*b)*x)+b*x+b+a)/a];
 199 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3])];
 200 [clairaut,0,0,0];
 201
 202 /*  430  */
 203 /* This equation presents a problem - runs forever */
 204 /* (pn_( 430 ),ans:contrib_ode(eqn: (a2*x+c_2)*('diff(y,x,1))^2+(b1*y+a1*x+c_1)*'diff(y,x,1)+b0*y+a0*x+c_0 ,y,x)); */
 205
 206 /*  431  */
 207 /* This equation presents a problem - crashes ode_contrib */
 208 /* (pn_( 431 ),ans:contrib_ode(eqn: x^2*('diff(y,x,1))^2-y^4+y^2 ,y,x)); */
 209
 210 /*  432  */
 211 /* This equation presents a problem - runs forever */
 212 /* (pn_( 432 ),ans:contrib_ode(eqn: (x*'diff(y,x,1)+a)^2-2*a*y+x^2 ,y,x)); */
 213
 214 /*  433  "Solution" changed in Jan 2007.  Don't worry yet. */
 215 /* (pn_(433),ans:contrib_ode(eqn:(x*'diff(y,x,1)+y+2*x)^2-4*(x*y+x^2+a),y,x));
 216 [-(3*'integrate((sqrt(x)*((sqrt(2)*sqrt(3)*y+2*sqrt(2)*sqrt(3)*x)*'integrate((%i*x*y+2*%i*a)*%e^-(log(x*y+x^2+a)/2)/(2*x*y+2*x^2+2*a),x)-2*sqrt(2)*sqrt(3)*sqrt(-x)*%e^(log(x)/2))+(-sqrt(2)*sqrt(3)*%e^(log(x)/2)*y-2*sqrt(2)*sqrt(3)*x*%e^(log(x)/2))*'integrate((%i*x*y+2*%i*a)*%e^-(log(x*y+x^2+a)/2)/(2*x*y+2*x^2+2*a),x)+2*sqrt(2)*sqrt(3)*sqrt(x)*sqrt(x*y+x^2+a)*'integrate((%i*x*y+2*%i*a)*%e^-(log(x*y+x^2+a)/2)/(2*x*y+2*x^2+2*a),x))/(6*sqrt(x)*sqrt(x*y+x^2+a)+sqrt(x)*(3*y+6*x)-3*%e^(log(x)/2)*y-6*x*%e^(log(x)/2)),y)-sqrt(2)*sqrt(3)*'integrate((%i*y+2*%i*x)*%e^-(log(x*y+x^2+a)/2),x)-2*sqrt(2)*sqrt(3)*%i*x)/3=%c];
 217 method;
 218 lie; */
 219
 220 /*  434  */
 221 (pn_(434),ans:contrib_ode(eqn:x^2*('diff(y,x,1))^2-2*x*y*'diff(y,x,1)-x^2,y,x));
 222 [x*(2*%c*y-x^2+%c^2)=0];
 223 solve(ans[1],y);
 224 [y=(x^2-%c^2)/(2*%c)];
 225 [method,ode_check(eqn,%[1])];
 226 [lagrange,0];
 227
 228 /*  435  */
 229 /* This equation presents a problem - crashes ode_contrib */
 230 /* (pn_( 435 ),ans:contrib_ode(eqn: x^2*('diff(y,x,1))^2-2*x*y*'diff(y,x,1)+y*(y+1)-x ,y,x)); */
 231
 232 /*  436  */
 233 assume(y>0);
 234 [y>0];
 235 (pn_(436),ans:contrib_ode(eqn:x^2*('diff(y,x,1))^2-2*x*y*'diff(y,x,1)+(1-x^2)*y^2-x^4,y,x));
 236 [asinh(y/abs(x))+x=%c];
 237 [method,ode_check(eqn,ans[1])];
 238 [lie,0];
 239 forget(y>0);
 240 [y>0];
 241
 242 /*  437  */
 243 (pn_(437),ans:contrib_ode(eqn:x^2*('diff(y,x,1))^2-(2*x*y+a)*'diff(y,x,1)+y^2,y,x));
 244 [y=%c*x-sqrt(%c*a),y=%c*x+sqrt(%c*a),y=-a/(4*x)];
 245 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3])];
 246 [clairaut,0,0,0];
 247
 248 /*  438  */
 249 (pn_(438),ans:contrib_ode(eqn:x^2*('diff(y,x,1))^2+3*x*y*'diff(y,x,1)+2*y^2,y,x));
 250 [y=%c/x,y=%c/x^2];
 251 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 252 [factor,0,0];
 253
 254 /*  439  CHECKME: Complex solution */
 255 assume(%t>0);
 256 [%t>0];
 257 (pn_(439),ans:contrib_ode(eqn:x^2*('diff(y,x,1))^2+3*x*y*'diff(y,x,1)+3*y^2,y,x));
 258 [[x = %c*%e^-(((sqrt(3)*%i+3)*log(%t))/(sqrt(3)*%i+9)),
 259   y = -((sqrt(3)*%i+3)*%t*x)/6],
 260  [x = %c*%e^-(((sqrt(3)*%i-3)*log(%t))/(sqrt(3)*%i-9)),
 261   y = ((sqrt(3)*%i-3)*%t*x)/6]]$
 262 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 263 [lagrange,0,0];
 264 forget(%t>0);
 265 [%t>0];
 266
 267 /*  440  */
 268 (pn_(440),ans:contrib_ode(eqn:x^2*('diff(y,x,1))^2+4*x*y*'diff(y,x,1)-5*y^2,y,x));
 269 [y=%c*x,y=%c/x^5];
 270 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 271 [factor,0,0];
 272
 273 /*  441  */
 274 (pn_(441),ans:contrib_ode(eqn:x^2*('diff(y,x,1))^2-4*x*(y+2)*'diff(y,x,1)+4*y*(y+2),y,x),0);
 275 0$
 276 /*
 277 [-(4*%i*sqrt(y+2)-2^(3/2)*%i*y-2^(5/2)*%i)/(sqrt(3)*x*sqrt(y+2))=%c]$
 278  */
 279 [method,ode_check(eqn,ans[1])];
 280 [lie,0];
 281
 282 /*  442  */
 283 (pn_(442),ans:contrib_ode(eqn:x^2*('diff(y,x,1))^2+(x^2*y-2*x*y+x^3)*'diff(y,x,1)+(1-x)*(y^2-x^2*y),y,x));
 284 [y=(%c-x)*x,y=%c*x*%e^-x];
 285 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 286 [factor,0,0];
 287
 288 /*  443  Just accept solution for now */
 289 (pn_(443),ans:contrib_ode(eqn:x*(x*'diff(y,x,1)-y)^2-'diff(y,x,1),y,x));
 290 ['integrate((%e^('integrate(1/x,x))*sqrt(4*x^2*y+1)
 291                                           *'integrate(
 292                                            (sqrt(4*x^2*y+1)
 293                                             *(12*x^6*y^2+12*x^4*y+2*x^2)
 294                                             -32*x^6*y^2-16*x^4*y-2*x^2)
 295                                             /((-216*x^8*%e^('integrate(1/x,x))
 296                                                    *y^4)
 297                                              +sqrt(4*x^2*y+1)
 298                                               *(108*x^6*%e^('integrate(1/x,x))
 299                                                    *y^3
 300                                                +67*x^4*%e^('integrate(1/x,x))
 301                                                   *y^2
 302                                                +14*x^2*%e^('integrate(1/x,x))
 303                                                   *y+%e^'integrate(1/x,x))
 304                                              -234*x^6*%e^('integrate(1/x,x))
 305                                                  *y^3
 306                                              -93*x^4*%e^('integrate(1/x,x))
 307                                                 *y^2
 308                                              -16*x^2*%e^('integrate(1/x,x))*y
 309                                              -%e^'integrate(1/x,x)),x)
 310                     +((-6*x^2*%e^('integrate(1/x,x))*y)-%e^'integrate(1/x,x))
 311                      *'integrate((sqrt(4*x^2*y+1)*(12*x^6*y^2+12*x^4*y+2*x^2)
 312                                   -32*x^6*y^2-16*x^4*y-2*x^2)
 313                                   /((-216*x^8*%e^('integrate(1/x,x))*y^4)
 314                                    +sqrt(4*x^2*y+1)
 315                                     *(108*x^6*%e^('integrate(1/x,x))*y^3
 316                                      +67*x^4*%e^('integrate(1/x,x))*y^2
 317                                      +14*x^2*%e^('integrate(1/x,x))*y
 318                                      +%e^'integrate(1/x,x))
 319                                    -234*x^6*%e^('integrate(1/x,x))*y^3
 320                                    -93*x^4*%e^('integrate(1/x,x))*y^2
 321                                    -16*x^2*%e^('integrate(1/x,x))*y
 322                                    -%e^'integrate(1/x,x)),x)+2*x^3)
 323                     /(%e^('integrate(1/x,x))*sqrt(4*x^2*y+1)
 324                      -6*x^2*%e^('integrate(1/x,x))*y-%e^'integrate(1/x,x)),y)
 325          +'integrate((sqrt(4*x^2*y+1)-2*x^2*y-1)
 326                       /(%e^('integrate(1/x,x))*sqrt(4*x^2*y+1)
 327                        -6*x^2*%e^('integrate(1/x,x))*y-%e^'integrate(1/x,x)),
 328                      x)
 329           = %c]$
 330 method;
 331 lie$
 332
 333 /*  444  */
 334 /* This equation presents a problem - runs forever */
 335 /* (pn_( 444 ),ans:contrib_ode(eqn: x^2*('diff(y,x,1))^2-y*(y-2*x)*'diff(y,x,1)+y^2 ,y,x)); */
 336
 337 /*  445  */
 338 (pn_(445),ans:contrib_ode(eqn:x^2*('diff(y,x,1))^2+(a*x^2*y^3+b)*'diff(y,x,1)+a*b*y^3,y,x));
 339 [y=b/x+%c,1/(2*a*y^2)=x+%c];
 340 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 341 [factor,0,0];
 342
 343 /*  446  */
 344 (pn_(446),ans:contrib_ode(eqn:(x^2+1)*('diff(y,x,1))^2-2*x*y*'diff(y,x,1)+y^2-1,y,x));
 345 [y=%c*x-sqrt(1-%c^2),y=%c*x+sqrt(1-%c^2),y=-sqrt(x^2+1),y=sqrt(x^2+1)];
 346 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3]),ode_check(eqn,ans[4])];
 347 [clairaut,0,0,0,0];
 348
 349 /*  447  */
 350 (pn_(447),ans:contrib_ode(eqn:(x^2-1)*('diff(y,x,1))^2-1,y,x));
 351 [-y-log(2*sqrt(x^2-1)+2*x)=%c];
 352 [method,ode_check(eqn,ans[1])];
 353 [lie,0];
 354
 355 /*  448  */
 356 /* This equation presents a problem - crashes ode_contrib */
 357 /* (pn_( 448 ),ans:contrib_ode(eqn: (x^2-1)*('diff(y,x,1))^2-y^2+1 ,y,x)); */
 358
 359 /*  449  */
 360 (pn_(449),ans:contrib_ode(eqn:(x^2-a^2)*('diff(y,x,1))^2+2*x*y*'diff(y,x,1)+y^2,y,x));
 361 [y=%c/(x-a),y=%c/(x+a)];
 362 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 363 [factor,0,0];
 364
 365 /*  450  */
 366 (pn_(450),ans:contrib_ode(eqn:(x^2-a^2)*('diff(y,x,1))^2-2*x*y*'diff(y,x,1)-x^2,y,x));
 367 [log(sqrt(y^2+x^2-a^2)-y)=%c];
 368 [method,ode_check(eqn,ans[1])];
 369 [lie,0];
 370
 371 /*  451  */
 372 (pn_(451),ans:contrib_ode(eqn:(x^2+a)*('diff(y,x,1))^2-2*x*y*'diff(y,x,1)+y^2+b,y,x));
 373 [y=%c*x-sqrt(-b-%c^2*a),y=%c*x+sqrt(-b-%c^2*a),y=-sqrt(-b*x^2/a-b),y=sqrt(-b*x^2/a-b)];
 374 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3]),ode_check(eqn,ans[4])];
 375 [clairaut,0,0,0,0];
 376
 377 /*  452  */
 378 /* This equation presents a problem - crashes ode_contrib */
 379 /* (pn_( 452 ),ans:contrib_ode(eqn: (2*x^2+1)*('diff(y,x,1))^2+(y^2+2*x*y+x^2+2)*'diff(y,x,1)+2*y^2+1 ,y,x)); */
 380
 381 /*  453   Not sure about this one - ode_check() chokes */
 382 (pn_( 453 ),ans:contrib_ode(eqn:(a^2-1)*x^2*('diff(y,x,1))^2+2*x*y*'diff(y,x,1)-y^2+a^2*x^2,y,x));
 383 ['integrate((a*sqrt(y^2+(1-a^2)*x^2)+y)/(a*%e^('integrate(1/x,x))*sqrt(y^2+(1-a^2)*x^2)+a^2*%e^('integrate(1/x,x))*y),x)-'integrate((a*%e^('integrate(1/x,x))*sqrt(y^2+(1-a^2)*x^2)*'integrate(((a^5-2*a^3+a)*x^2*y*sqrt(y^2+(1-a^2)*x^2)+(a^4-2*a^2+1)*x^2*y^2+(-a^6+3*a^4-3*a^2+1)*x^4)/((a^4+3*a^2)*%e^('integrate(1/x,x))*y^5+sqrt(y^2+(1-a^2)*x^2)*((3*a^3+a)*%e^('integrate(1/x,x))*y^4+(-3*a^5+a^3+2*a)*x^2*%e^('integrate(1/x,x))*y^2+(a^5-2*a^3+a)*x^4*%e^'integrate(1/x,x))+(-a^6-5*a^4+6*a^2)*x^2*%e^('integrate(1/x,x))*y^3+(3*a^6-6*a^4+3*a^2)*x^4*%e^('integrate(1/x,x))*y),x)+a^2*%e^('integrate(1/x,x))*y*'integrate(((a^5-2*a^3+a)*x^2*y*sqrt(y^2+(1-a^2)*x^2)+(a^4-2*a^2+1)*x^2*y^2+(-a^6+3*a^4-3*a^2+1)*x^4)/((a^4+3*a^2)*%e^('integrate(1/x,x))*y^5+sqrt(y^2+(1-a^2)*x^2)*((3*a^3+a)*%e^('integrate(1/x,x))*y^4+(-3*a^5+a^3+2*a)*x^2*%e^('integrate(1/x,x))*y^2+(a^5-2*a^3+a)*x^4*%e^'integrate(1/x,x))+(-a^6-5*a^4+6*a^2)*x^2*%e^('integrate(1/x,x))*y^3+(3*a^6-6*a^4+3*a^2)*x^4*%e^('integrate(1/x,x))*y),x)+(1-a^2)*x)/(a*%e^('integrate(1/x,x))*sqrt(y^2+(1-a^2)*x^2)+a^2*%e^('integrate(1/x,x))*y),y) = %c];
 384 method;
 385 lie;
 386
 387 /*  454  */
 388 assume(a>1,y>0);
 389 [a>1,y>0];
 390 (pn_(454),ans:contrib_ode(eqn:a*x^2*('diff(y,x,1))^2-2*a*x*y*'diff(y,x,1)+y^2-(a-1)*a*x^2,y,x));
 391 [(a*asinh(y/(sqrt(a)*abs(x)))+sqrt(a-1)*sqrt(a)*log(x))/(sqrt(a-1)*sqrt(a))=%c];
 392 [method,ode_check(eqn,ans[1])];
 393 [lie,0];
 394 forget(a>1,y>0);
 395 [a>1,y>0];
 396
 397 /*  455  */
 398 assume(a>0);
 399 [a>0];
 400 (pn_(455),ans:contrib_ode(eqn: x^3*('diff(y,x,1))^2+x^2*y*'diff(y,x,1)+a ,y,x));
 401  [(abs(y)*'integrate((2*x*abs(y)^3-2*x*y^2*abs(y))/(sqrt(x^2*y^2-4*a*x)*abs(y)^3+x*y^4-2*a*y^2),y)+y*log(2*sqrt(x^2*y^2-4*a*x)*abs(y)+2*x*y^2-4*a)+log(x)*abs(y))/abs(y)=%c];
 402 method;
 403 lie;
 404 forget(a>0);
 405 [a>0];
 406
 407 /*  456  */
 408 assume(x>1);
 409 [x>1];
 410 (pn_(456),ans:contrib_ode(eqn: x*(x^2-1)*('diff(y,x,1))^2+2*(1-x^2)*y*'diff(y,x,1)+x*y^2-x ,y,x));
 411 [(sqrt(x-1)*sqrt(x+1)*('integrate((sqrt(x^2-y^2)-sqrt(x^2-1)*y)/(sqrt(x-1)*x*sqrt(x+1)*sqrt(x^2-y^2)),x)-'integrate('integrate(sqrt(x-1)*x*sqrt(x+1)*sqrt(x^2-1)/(sqrt(x^2-y^2)*((x^2-1)*y^2-x^4+x^2)),x),y))+sqrt(x^2-1)*asin(y/x))/(sqrt(x-1)*sqrt(x+1))=%c];
 412 method;
 413 lie;
 414 forget(x>1);
 415 [x>1];
 416
 417 /*  457 - "solution" changed Jan 2007.  Don't care at present. */
 418 /*
 419 (pn_(457),ans:contrib_ode(eqn:x^4*('diff(y,x,1))^2-x*'diff(y,x,1)-y,y,x));
 420 [(2*%i*x^2*sqrt(4*x^2*y+1)*'integrate((abs(x)*sqrt(4*x^2*y+1)-abs(x))/(x^3*sqrt(4*x^2*y+1)),x)+8*%i*x^2*abs(x)*y+2*%i*'integrate(abs(x)/x,x)+2*%i*abs(x))/(sqrt(2)*sqrt(3)*x^2*sqrt(4*x^2*y+1))=%c];
 421 method;
 422 lie;
 423 */
 424
 425 /*  458 - "solution" changed Jan 2007.  Don't care at present. */
 426 /*
 427 assume(a>0);
 428 [a>0];
 429 (pn_(458),ans:contrib_ode(eqn:x^2*(x^2-a^2)*('diff(y,x,1))^2-1,y,x));
 430 [-(a*y-asin(a/abs(x)))/a=%c];
 431 [method,ode_check(eqn,ans[1])];
 432 [lie,0];
 433 forget(a>0);
 434 [a>0];
 435 */
 436
 437 /*  459  */
 438 /* "solution" has changed - Jan 2007.  Don't worry at present
 439 assume(x>0);
 440 [x>0];
 441 (pn_(459),ans:contrib_ode(eqn:%e^-(2*x)*('diff(y,x,1))^2-('diff(y,x,1)-1)^2+%e^-(2*y),y,x));
 442 [(2*'integrate('integrate(%e^(-log(%e^(2*y)+%e^(2*x)-1)/2+y+2*x)/(%e^(2*y)+%e^(2*x)-1),x),y)-2*'integrate(%e^(-log(%e^(2*y)+%e^(2*x)-1)/2+y+2*x)/(%e^(2*x)-1),x)+2*asinh(%e^y/sqrt(%e^(2*x)-1))-log(%e^x+1)+log(%e^x-1))/2=%c];
 443 method;
 444 lie;
 445 forget(x>0);
 446 [x>0];
 447 */
 448
 449 /*  460  */
 450 (pn_(460),ans:contrib_ode(eqn:cos(x)^4*(('diff(y,x,1))^2+y^2)-a^2,y,x));
 451 false;
 452
 453 /*  461  */
 454 /* This equation too general - E S Cheb-Terrab and T Kolokolnikov */
 455 /* (pn_( 461 ),ans:contrib_ode(eqn: A(x)*('diff(y,x,1))^2+2*B(x)*y*'diff(y,x,1)+2*DD(x)*'diff(y,x,1)+C(x)*y^2+2*EE(x)*y+F(x) ,y,x)); */
 456
 457 /*  462  */
 458 (pn_(462),ans:contrib_ode(eqn:y*('diff(y,x,1))^2-1,y,x));
 459 [4*y^3-9*x^2+18*%c*x-9*%c^2=0];
 460  solve(ans[1],y);
 461 [y=(sqrt(3)*9^(1/3)*%i-9^(1/3))*(x-%c)^(2/3)/(2*4^(1/3)),y=-((sqrt(3)*9^(1/3)*%i+9^(1/3))*(x-%c)^(2/3)/(2*4^(1/3))),y=9^(1/3)*(x-%c)^(2/3)/4^(1/3)];
 462 [method,ode_check(eqn,%[3])];
 463 [lagrange,0];
 464
 465 /*  463  */
 466 /* This equation presents a problem - crashes ode_contrib */
 467 /* (pn_( 463 ),ans:contrib_ode(eqn: y*('diff(y,x,1))^2-%e^(2*x),y,x)); */
 468
 469 /*  464  */
 470 (pn_(464),ans:contrib_ode(eqn: y*('diff(y,x,1))^2+2*x*'diff(y,x,1)-y,y,x));
 471 [[x=%c*%e^(2*(log(%t+1)/2-log(%t)+log(%t-1)/2)),y=-2*%t*x/(%t^2-1)]];
 472 [method,ode_check(eqn,ans[1])];
 473 [lagrange,0];
 474
 475 /*  465  */
 476 (pn_(465),ans:contrib_ode(eqn:y*('diff(y,x,1))^2+2*x*'diff(y,x,1)-9*y,y,x));
 477 [[x=%c*(%t-3)*(%t+3)*%e^(2*log(%t)/7-8*log(%t^2-7)/7),y=-2*%t*x/(%t^2-9)]];
 478 [method,ode_check(eqn,ans[1])];
 479 [lagrange,0];
 480
 481 /*  466  */
 482 (pn_(466),ans:contrib_ode(eqn:y*('diff(y,x,1))^2-2*x*'diff(y,x,1)+y,y,x));
 483 [[x=%c*%e^-(2*(log(%t)-log(%t^2+1)/2)),y=2*%t*x/(%t^2+1)]];
 484 [method,ode_check(eqn,ans[1])];
 485 [lagrange,0];
 486
 487 /*  467  */
 488 (pn_(467),ans:contrib_ode(eqn:y*('diff(y,x,1))^2-4*x*'diff(y,x,1)+y,y,x));
 489 [[x=%c*(%t^2+1)*%e^(-log(%t^2-3)/3-4*log(%t)/3),y=4*%t*x/(%t^2+1)]];
 490 [method,ode_check(eqn,ans[1])];
 491 [lagrange,0];
 492
 493 /*  468  */
 494 (pn_(468),ans:contrib_ode(eqn:y*('diff(y,x,1))^2-4*a^2*x*'diff(y,x,1)+a^2*y,y,x));
 495 [[x=%c*(a^2+%t^2)*%e^(-log(%t^2-3*a^2)/3-4*log(%t)/3),y=4*%t*a^2*x/(a^2+%t^2)]];
 496 [method,ode_check(eqn,ans[1])];
 497 [lagrange,0];
 498
 499 /*  469  */
 500 (pn_(469),ans:contrib_ode(eqn:y*('diff(y,x,1))^2+a*x*'diff(y,x,1)+b*y,y,x));
 501 [[x=%c*(b+%t^2)*%e^(-((2*b+a)*log(b+a+%t^2)/(2*b+2*a))-log(%t)*a/(b+a)),y=-(%t*a*x/(b+%t^2))]];
 502 [method,ode_check(eqn,ans[1])];
 503 [lagrange,0];
 504
 505 /*  470  */
 506 (pn_(470),ans:contrib_ode(eqn:y*('diff(y,x,1))^2+x^3*'diff(y,x,1)-x^2*y,y,x));
 507 [(log((x^2*sqrt(4*y^2+x^4)+2*y^2+x^4)/x^2)+2*log(x))/4=%c];
 508 [method,ode_check(eqn,ans[1])];
 509 [lie,0];
 510
 511 /*  471  */
 512 (pn_(471),ans:contrib_ode(eqn:y*('diff(y,x,1))^2-(y-x)*'diff(y,x,1)-x,y,x));
 513 [y=x+%c,-y^2/2=x^2/2+%c];
 514 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 515 [factor,0,0];
 516
 517 /*  472  */
 518 (pn_(472),ans:contrib_ode(eqn:(y+x)*('diff(y,x,1))^2+2*x*'diff(y,x,1)-y,y,x));
 519 [[x=%c*%e^(2*(log(%t+1)/2-log(%t)+log(%t-1)/2)),y=-((%t^2+2*%t)*x/(%t^2-1))]];
 520 [method,ode_check(eqn,ans[1])];
 521 [lagrange,0];
 522
 523 /*  473  */
 524 /* This equation presents a problem - runs forever */
 525 /* (pn_( 473 ),ans:contrib_ode(eqn: (y-2*x)*('diff(y,x,1))^2-2*(x-1)*'diff(y,x,1)+y-2 ,y,x)); */
 526
 527 /*  474  */
 528 /* This equation presents a problem - runs forever */
 529 /* (pn_( 474 ),ans:contrib_ode(eqn: 2*y*('diff(y,x,1))^2-(4*x-5)*'diff(y,x,1)+2*y ,y,x)); */
 530
 531 /*  475  */
 532 (pn_(475),ans:contrib_ode(eqn: 4*y*('diff(y,x,1))^2+2*x*'diff(y,x,1)-y ,y,x));
 533 [[x=%c*%e^(2*(log(2*%t+1)/2+log(2*%t-1)/2-log(%t))),y=-2*%t*x/(4*%t^2-1)]];
 534 [method,ode_check(eqn,ans[1])];
 535 [lagrange,0];
 536
 537 /*  476  */
 538 (pn_(476),ans:contrib_ode(eqn:9*y*('diff(y,x,1))^2+4*x^3*'diff(y,x,1)-4*x^2*y,y,x));
 539 [(log((2*x^2*sqrt(9*y^2+x^4)+9*y^2+2*x^4)/(2*x^2))+2*log(x))/4=%c];
 540 [method,ode_check(eqn,ans[1])];
 541 [lie,0];
 542
 543 /*  477  */
 544 /* This equation presents a problem - runs forever */
 545 /* (pn_( 477 ),ans:contrib_ode(eqn: a*y*('diff(y,x,1))^2+(2*x-b)*'diff(y,x,1)-y ,y,x)); */
 546
 547 /*  478  */
 548 (pn_(478),ans:contrib_ode(eqn:(a*y+b)*(('diff(y,x,1))^2+1)-c,y,x));
 549 [[x=%c-(2*(%t/(2*%t^2*a+2*a)+atan(%t)/(2*a)))*c,y=-((-c+%t^2*b+b)/(%t^2*a+a))]];
 550 [method,ode_check(eqn,ans[1])];
 551 [lagrange,0];
 552
 553 /*  479  */
 554 /* This equation presents a problem - runs forever */
 555 /* (pn_( 479 ),ans:contrib_ode(eqn: (b2*y+a2*x+c_2)*('diff(y,x,1))^2+(b1*y+a1*x+c_1)*'diff(y,x,1)+b0*y+a0*x+c_0 ,y,x)); */
 556
 557 /*  480  */
 558 (pn_(480),ans:contrib_ode(eqn:(a*y-x^2)*('diff(y,x,1))^2+2*x*y*('diff(y,x,1))^2-y^2,y,x));
 559 false;
 560
 561 /*  481  */
 562 (pn_(481),ans:contrib_ode(eqn:x*y*('diff(y,x,1))^2+(y^2+x^2)*'diff(y,x,1)+x*y,y,x));
 563 [y=%c/x,-y^2/2=x^2/2+%c];
 564 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 565 [factor,0,0];
 566
 567 /*  482  */
 568 /* This equation presents a problem - runs forever */
 569 /* (pn_( 482 ),ans:contrib_ode(eqn: x*y*('diff(y,x,1))^2+(-y^2+x^2+a)*'diff(y,x,1)-x*y ,y,x)); */
 570
 571 /*  483  */
 572 /* This equation presents a problem - runs forever */
 573 /* (pn_( 483 ),ans:contrib_ode(eqn: (2*x*y-x^2)*('diff(y,x,1))^2+2*x*y*'diff(y,x,1)-y^2+2*x*y ,y,x)); */
 574
 575 /*  484  */
 576 /* This equation presents a problem - runs forever */
 577 /* (pn_( 484 ),ans:contrib_ode(eqn: (2*x*y-x^2)*('diff(y,x,1))^2-6*x*y*'diff(y,x,1)-y^2+2*x*y ,y,x)); */
 578
 579 /*  485  */
 580 /* This equation presents a problem - runs forever */
 581 /* (pn_( 485 ),ans:contrib_ode(eqn: a*x*y*('diff(y,x,1))^2-(a*y^2+b*x^2+c)*'diff(y,x,1)+b*x*y ,y,x)); */
 582
 583 /*  486  */
 584 (pn_(486),ans:contrib_ode(eqn: y^2*('diff(y,x,1))^2+y^2-a^2 ,y,x));
 585 [[x=%t*a/sqrt(%t^2+1)+%c,y = -a/sqrt(%t^2+1)],[x=%c-%t*a/sqrt(%t^2+1),y = a/sqrt(%t^2+1)]];
 586 [method,ode_check(eqn,ans[1])];
 587 [lagrange,0];
 588
 589 /*  487 - "solution" changed Jan 2007.  Don't care at present. */
 590 /*
 591 (pn_(487),ans:contrib_ode(eqn:y^2*('diff(y,x,1))^2-6*x^3*'diff(y,x,1)+4*x^2*y,y,x));
 592 [-(4*'integrate((sqrt(9*x^4-4*y^3)*(27*%i*x^4*y^2*%e^(log(4*y^3-9*x^4)/2)+36*x^2*y^5-81*x^6*y^2)+(81*%i*x^6*y^2-36*%i*x^2*y^5)*%e^(log(4*y^3-9*x^4)/2)+108*x^4*y^5-243*x^8*y^2)/(sqrt(9*x^4-4*y^3)*((72*%i*x^4*y^3-162*%i*x^8)*%e^(log(4*y^3-9*x^4)/2)+48*x^2*y^6-324*x^6*y^3+486*x^10)+(96*%i*x^2*y^6-432*%i*x^6*y^3+486*%i*x^10)*%e^(log(4*y^3-9*x^4)/2)+64*y^9-576*x^4*y^6+1620*x^8*y^3-1458*x^12),y)-log((3*x^2*%e^(log(4*y^3-9*x^4)/2)-2*%i*y^3+9*%i*x^4)/(3*x^2))-2*log(x))/4=%c];
 593 method;
 594 lie;
 595 */
 596
 597 /*  488  */
 598 /* This equation presents a problem - runs forever */
 599 /* (pn_( 488 ),ans:contrib_ode(eqn: y^2*('diff(y,x,1))^2-4*a*y*'diff(y,x,1)+y^2-4*a*x+4*a^2 ,y,x)); */
 600
 601 /*  489  */
 602 /* This equation presents a problem - runs forever */
 603 /* (pn_( 489 ),ans:contrib_ode(eqn: y^2*('diff(y,x,1))^2+2*x*y*'diff(y,x,1)+a*y^2+b*x+c ,y,x)); */
 604
 605 /*  490  */
 606 /* This equation presents a problem - runs forever */
 607 /* (pn_( 490 ),ans:contrib_ode(eqn: y^2*('diff(y,x,1))^2-2*x*y*'diff(y,x,1)+2*y^2-x^2+a ,y,x)); */
 608
 609 /*  491  */
 610 /* This equation presents a problem - runs forever */
 611 /* (pn_( 491 ),ans:contrib_ode(eqn: y^2*('diff(y,x,1))^2+2*a*x*y*'diff(y,x,1)+(1-a)*y^2+a*x^2+(a-1)*b ,y,x)); */
 612
 613 /*  492  */
 614 (pn_( 492 ),ans:contrib_ode(eqn: (y^2-a^2)*('diff(y,x,1))^2+y^2 ,y,x));
 615 [[x=%c-('integrate(1/(sqrt(%t^2+1)*(%t^3+%t)),%t))*a,y=-%t*a/sqrt(%t^2+1)],[x=('integrate(1/(sqrt(%t^2+1)*(%t^3+%t)),%t))*a+%c,y = %t*a/sqrt(%t^2+1)]];
 616 [method,ode_check(eqn,ans[1])];
 617 [lagrange,0];
 618
 619 /*  493  */
 620 /* This equation presents a problem - runs forever */
 621 /* (pn_( 493 ),ans:contrib_ode(eqn: (y^2-2*a*x+a^2)*('diff(y,x,1))^2+2*a*y*'diff(y,x,1)+y^2 ,y,x)); */
 622
 623 /*  494  */
 624 /* This equation presents a problem - runs forever */
 625 /* (pn_( 494 ),ans:contrib_ode(eqn: (y^2-a^2*x^2)*('diff(y,x,1))^2+2*x*y*'diff(y,x,1)+(1-a^2)*x^2 ,y,x)); */
 626
 627 /*  495  */
 628 /* This equation presents a problem - crashes ode_contrib */
 629 /* (pn_( 495 ),ans:contrib_ode(eqn: (y^2+(1-a)*x^2)*('diff(y,x,1))^2+2*a*x*y*'diff(y,x,1)+(1-a)*y^2+x^2 ,y,x)); */
 630
 631 /*  496  */
 632 /* This equation presents a problem - crashes ode_contrib */
 633 /* (pn_( 496 ),ans:contrib_ode(eqn: (y-x)^2*(('diff(y,x,1))^2+1)-a^2*('diff(y,x,1)+1)^2 ,y,x)); */
 634
 635 /*  497  */
 636 /* This equation presents a problem - runs forever */
 637 /* (pn_( 497 ),ans:contrib_ode(eqn: 3*y^2*('diff(y,x,1))^2-2*x*y*'diff(y,x,1)+4*y^2-x^2 ,y,x)); */
 638
 639 /*  498  */
 640 (pn_(498),ans:contrib_ode(eqn:(3*y-2)*('diff(y,x,1))^2+4*(y-1),y,x));
 641 [[x=%c-8*(atan(3*%t/(2*sqrt(3)))/(16*sqrt(3))+%t/(24*%t^2+32)),y=(2*%t^2+4)/(3*%t^2+4)]];
 642 [method,ode_check(eqn,ans[1])];
 643 [lagrange,0];
 644
 645 /*  499  */
 646 /* This equation presents a problem - runs forever */
 647 /* (pn_( 499 ),ans:contrib_ode(eqn: (1-a^2)*y^2*('diff(y,x,1))^2-2*a^2*x*y*'diff(y,x,1)+y^2-a^2*x^2 ,y,x)); */
 648
 649 /*  500  */
 650 /* This equation presents a problem - runs forever */
 651 /* (pn_( 500 ),ans:contrib_ode(eqn: (a-b)*y^2*('diff(y,x,1))^2-2*b*x*y*'diff(y,x,1)+a*y^2-b*x^2-a*b ,y,x)); */
 652