share/contrib/diffequations/tests/rtestode_kamke_1_1.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Equations 1-100
   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 /*   1 */
  22 (pn_(1),ans:contrib_ode(eqn:'diff(y,x)-(a4*x^4+a3*x^3+a2*x^2+a1*x+a0)^(-1/2),y,x));
  23 [y = 'integrate(1/sqrt(a4*x^4+a3*x^3+a2*x^2+a1*x+a0),x)+%c];
  24 [method,ode_check(eqn,ans[1])];
  25 [linear,0];
  26
  27 /* 2 */
  28 (pn_(2),ans:contrib_ode(eqn:'diff(y,x)+a*y-c*exp(b*x),y,x));
  29 [y = %e^-(a*x)*(c*%e^(b*x+a*x)/(b+a)+%c)];
  30 [method,ode_check(eqn,ans[1])];
  31 [linear,0];
  32
  33 /* 3 */
  34 (pn_(3),ans:contrib_ode(eqn:'diff(y,x)+a*y-b*sin(c*x),y,x));
  35 [y=%e^-(a*x)*(b*%e^(a*x)*(a*sin(c*x)-c*cos(c*x))/(c^2+a^2)+%c)];
  36 [method,ode_check(eqn,ans[1])];
  37 [linear,0];
  38
  39 /* 4 */
  40 (pn_(4),ans:contrib_ode(eqn:'diff(y,x)+2*x*y-x*exp(-x^2),y,x));
  41 [y=(x^2/2+%c)*%e^-x^2];
  42 [method,ode_check(eqn,ans[1])];
  43 [linear,0];
  44
  45 /* 5 */
  46 (pn_(5),ans:contrib_ode(eqn:'diff(y,x)+y*cos(x)-exp(2*x),y,x));
  47 [y=%e^-sin(x)*('integrate(%e^(sin(x)+2*x),x)+%c)];
  48 [method,ode_check(eqn,ans[1])];
  49 [linear,0];
  50
  51 /* 6 */
  52 (pn_(6),ans:contrib_ode(eqn:'diff(y,x)+y*cos(x)-sin(2*x)/2,y,x));
  53 [y=%e^-sin(x)*((sin(x)-1)*%e^sin(x)+%c)];
  54 [method,ode_check(eqn,ans[1])];
  55 [linear,0];
  56
  57 /* 7 */
  58 (pn_(7),ans:contrib_ode(eqn:'diff(y,x)+y*cos(x)-exp(-sin(x)),y,x));
  59 [y = (x+%c)*%e^-sin(x)];
  60 [method,ode_check(eqn,ans[1])];
  61 [linear,0];
  62
  63 /* 8 */
  64 (pn_(8),ans:contrib_ode(eqn: 'diff(y,x) + y*tan(x) - sin(2*x),y,x));
  65 [y = (%c-2*cos(x))/sec(x)];
  66 [method,ode_check(eqn,ans[1])];
  67 [linear,0];
  68
  69 /* 9 */
  70 (pn_(9),ans:contrib_ode(eqn:'diff(y,x)-(sin(log(x))+cos(log(x))+a)*y,y,x));
  71 [y=%c*%e^(x*(sin(log(x))+cos(log(x)))/2+x*(sin(log(x))-cos(log(x)))/2+a*x)];
  72 [method,ode_check(eqn,ans[1])];
  73 [linear,0];
  74
  75 /* 10 */
  76 (pn_(10),ans:contrib_ode(eqn: 'diff(y,x) + 'diff(f(x),x)*y - f(x)*'diff(f(x),x),y,x));
  77 [y=%e^-f(x)*('integrate(f(x)*%e^f(x)*'diff(f(x),x,1),x)+%c)];
  78 [method,ode_check(eqn,ans[1])];
  79 [linear,0];
  80
  81 /* 11 */
  82 (pn_(11),ans:contrib_ode(eqn: 'diff(y,x)  + f(x)*y - g(x),y,x));
  83 [y = %e^-('integrate(f(x),x))*('integrate(g(x)*%e^'integrate(f(x),x),x)+%c)];
  84 [method,ode_check(eqn,ans[1])];
  85 [linear,0];
  86
  87 /* 12 */
  88 (pn_(12),ans:contrib_ode(eqn:'diff(y,x) + y^2 - 1,y,x));
  89 [(log(y+1)-log(y-1))/2=x+%c];
  90 solve(map(exp,2*ans[1]),y);
  91 [y = (%e^(2*x+2*%c)+1)/(%e^(2*x+2*%c)-1)];
  92 [method,ode_check(eqn,%[1])];
  93 [separable,0];
  94
  95 /* 13 - Riccati eqn */
  96 (pn_(13),ans:contrib_ode(eqn:'diff(y,x) + y^2 - a*x - b,y,x),0);
  97 0$
  98 /*
  99 [y = -(((-bessel_y(1/3,2*%i*(a*x+b)^(3/2)/(3*a))*%c
 100         -bessel_j(1/3,2*%i*(a*x+b)^(3/2)/(3*a)))*a*sqrt(a*x+b)
 101    +%i*((bessel_y(4/3,2*%i*(a*x+b)^(3/2)/(3*a))
 102         -bessel_y(-(2/3),2*%i*(a*x+b)^(3/2)/(3*a)))*%c
 103         +bessel_j(4/3,2*%i*(a*x+b)^(3/2)/(3*a))
 104         -bessel_j(-(2/3),2*%i*(a*x+b)^(3/2)/(3*a)))*a^2*x^2
 105  +%i*((2*bessel_y(4/3,2*%i*(a*x+b)^(3/2)/(3*a))
 106       -2*bessel_y(-(2/3),2*%i*(a*x+b)^(3/2)/(3*a)))*%c
 107       +2*bessel_j(4/3,2*%i*(a*x+b)^(3/2)/(3*a))
 108       -2*bessel_j(-(2/3),2*%i*(a*x+b)^(3/2)/(3*a)))*a*b*x
 109    +%i*((bessel_y(4/3,2*%i*(a*x+b)^(3/2)/(3*a))
 110         -bessel_y(-(2/3),2*%i*(a*x+b)^(3/2)/(3*a)))*%c
 111         +bessel_j(4/3,2*%i*(a*x+b)^(3/2)/(3*a))
 112         -bessel_j(-(2/3),2*%i*(a*x+b)^(3/2)/(3*a)))*b^2)
 113     /((2*bessel_y(1/3,2*%i*(a*x+b)^(3/2)/(3*a))*%c
 114       +2*bessel_j(1/3,2*%i*(a*x+b)^(3/2)/(3*a)))*a*x*sqrt(a*x+b)
 115      +(2*bessel_y(1/3,2*%i*(a*x+b)^(3/2)/(3*a))*%c
 116       +2*bessel_j(1/3,2*%i*(a*x+b)^(3/2)/(3*a)))*b*sqrt(a*x+b)))];
 117  */
 118 /* FIXME: bind stack overflow when checking 2015-04-06
 119 [method,ode_check(eqn,ans[1])];
 120 [riccati,0];
 121 */
 122
 123 /* 14 - The constraints need some work
 124          want m # -2, rather than m > -2 */
 125 assume(m>-2,a>0);
 126 [m>-2,a>0];
 127 (pn_("14i"),ans:contrib_ode(eqn: 'diff(y,x) + y^2 + a*x^m,y,x),0);
 128 0$
 129 /*
 130 [y = -((sqrt(a)*(bessel_j((m+3)/(m+2),2*sqrt(a)*x^(m/2+1)/(m+2))
 131             -%c*bessel_j(-((m+3)/(m+2)),2*sqrt(a)*x^(m/2+1)/(m+2))
 132             +%c*bessel_j((m+1)/(m+2),2*sqrt(a)*x^(m/2+1)/(m+2))
 133                -bessel_j(-((m+1)/(m+2)),2*sqrt(a)*x^(m/2+1)/(m+2)))*x^(m/2+1)
 134                -bessel_j(1/(m+2),2*sqrt(a)*x^(m/2+1)/(m+2))
 135             -%c*bessel_j(-(1/(m+2)),2*sqrt(a)*x^(m/2+1)/(m+2)))
 136           /((2*bessel_j(1/(m+2),2*sqrt(a)*x^(m/2+1)/(m+2))
 137          +2*%c*bessel_j(-(1/(m+2)),2*sqrt(a)*x^(m/2+1)/(m+2)))*x))];
 138  */
 139 [method,ratsimp(ode_check(eqn,ans[1]))];
 140 [riccati,0];
 141 forget(m>-2,a>0);
 142 [m>-2,a>0];
 143
 144 assume(m>-2,a<0);
 145 [m>-2,a<0];
 146 (pn_("14ii"),ans:contrib_ode(eqn: 'diff(y,x) + y^2 + a*x^m,y,x),0);
 147 0$
 148 /*
 149  [y = -(sqrt(-a)*(%c*bessel_k((m+1)/(m+2),2*sqrt(-a)*x^(m/2+1)/(m+2))-bessel_i(-((m+1)/(m+2)),2*sqrt(-a)*x^(m/2+1)/(m+2)))*x^(m/2)/(%c*bessel_k(1/(m+2),2*sqrt(-a)*x^(m/2+1)/(m+2))+bessel_i(1/(m+2),2*sqrt(-a)*x^(m/2+1)/(m+2))))];
 150  */
 151 [method,ratsimp(ode_check(eqn,ans[1]))];
 152 [riccati,0];
 153 forget(m>-2,a<0);
 154 [m>-2,a<0];
 155
 156 (m:-2,assume(a > 1/4));
 157 [a>1/4];
 158 (pn_("14iii"),ans:contrib_ode(eqn: 'diff(y,x) + y^2 + a*x^m,y,x));
 159 [ x = %c*%e^-(2*atan((2*x*y-1)/sqrt(4*a-1))/sqrt(4*a-1))];
 160 solve(map(log,ans[1]/%c),y);
 161 [y = -((sqrt(4*a-1)*tan(sqrt(4*a-1)*log(x/%c)/2)-1)/(2*x))];
 162 [method,ode_check(eqn,%[1])];
 163 [genhom,0];
 164 (kill(m),forget(a > 1/4));
 165 [a>1/4];
 166
 167 (m:-2,assume(a < 1/4));
 168 [a < 1/4];
 169 (pn_("14iv"),ans:contrib_ode(eqn: 'diff(y,x) + y^2 + a*x^m,y,x));
 170 [x=%c*%e^-(log(-((-2*x*y+sqrt(1-4*a)+1)/(2*x*y+sqrt(1-4*a)-1)))/sqrt(1-4*a))];
 171 solve(map(log,ans[1]/%c),y);
 172 [y = ((sqrt(1-4*a)+1)*%e^(sqrt(1-4*a)*log(x/%c))+sqrt(1-4*a)-1)/(2*x*%e^(sqrt(1-4*a)*log(x/%c))-2*x)];
 173 [method,ode_check(eqn,%[1])];
 174 [genhom,0];
 175 (kill(m),forget(a < 1/4));
 176 [a < 1/4];
 177
 178 /* 15 This is a Riccati eqn */
 179 (pn_(15),ans:contrib_ode(eqn:'diff(y,x) + y^2 - 2*x^2*y + x^4 -2*x-1,y,x));
 180  [y = (x^2*(%c*%e^(2*x)+1)+%c*%e^(2*x)-1)/(%c*%e^(2*x)+1)];
 181 [method,ode_check(eqn,ans[1])];
 182 [riccati,0];
 183
 184 /* 16 This is a Riccati eqn */
 185 (pn_(16),ans:contrib_ode(eqn:'diff(y,x) + y^2 +(x*y-1)*f(x),y,x));
 186 [[y='diff(%u,x,1)/%u,-'diff(%u,x,1)*x*f(x)+%u*f(x)-'diff(%u,x,2)=0]];
 187 method;
 188 riccati;
 189
 190 /* 17 */
 191 (pn_(17),ans:contrib_ode(eqn:'diff(y,x) - y^2 -3*y + 4,y,x));
 192 [-((log(y+4)-log(y-1))/5) = x+%c];
 193 solve(map(exp,ans[1]*5),y);
 194 [y = -((4*%e^(5*x+5*%c)+1)/(%e^(5*x+5*%c)-1))];
 195 [method,ode_check(eqn,%[1])];
 196 [separable,0];
 197
 198 /* 18 */
 199 (pn_(18),ans:contrib_ode(eqn:'diff(y,x)-y^2-x*y-x+1,y,x),0);
 200 0$
 201 /*
 202 [y = -(%e^-(2*x)*(%e^(2*x)*(sqrt(2)*sqrt(%pi)*%c
 203                                          *erf((%i*x-2*%i)/sqrt(2))
 204                                  +sqrt(2)*%e^2*sqrt(%pi))
 205                        +2*%i*%c*%e^(x^2/2+2))
 206  /(sqrt(2)*sqrt(%pi)*%c*erf((%i*x-2*%i)/sqrt(2))+sqrt(2)*%e^2*sqrt(%pi)))];
 207  */
 208 [method,ode_check(eqn,ans[1])];
 209 [riccati,0];
 210
 211 /* 19 */
 212 (pn_(19),ans:contrib_ode(eqn:'diff(y,x) - (y + x)^2,y,x));
 213 [[x=%c-atan(sqrt(%t)),y=-x-sqrt(%t)],[x=atan(sqrt(%t))+%c,y=sqrt(%t)-x]];
 214 /*
 215 solve(ans[1][2],sqrt(%t));
 216 [sqrt(%t) = -y-x];
 217 ans[1][1],%;
 218 x=atan(y+x)+%c;
 219 ans[1]:first(solve(%,y));
 220 y = tan(x-%c)-x;
 221 solve(ans[2][2],sqrt(%t));
 222 [sqrt(%t) = y+x];
 223 ans[2][1],%;
 224 x = atan(y+x)+%c;
 225 ans[2]:first(solve(%,y));
 226 y = tan(x-%c)-x;
 227 */
 228 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 229 [lagrange,0,0];
 230
 231 /* 20 */
 232 (pn_(20),ans:contrib_ode(eqn:'diff(y,x)-y^2+(x^2+1)*y-2*x,y,x));
 233 [y = ((x^2+1)*'integrate(%e^(x^3/3+x),x)-%e^(x^3/3+x)+%c*x^2+%c)/('integrate(%e^(x^3/3+x),x)+%c)];
 234 [method,ode_check(eqn,ans[1])];
 235 [riccati,0];
 236
 237 /* 21 */
 238 (pn_(21),ans:contrib_ode(eqn:'diff(y,x)-y^2+y*sin(x)-cos(x),y,x));
 239  [y = (%e^cos(x)*sin(x)*'integrate(%e^-cos(x),x)+%c*%e^cos(x)*sin(x)-1)/(%e^cos(x)*'integrate(%e^-cos(x),x)+%c*%e^cos(x))];
 240 [method,ode_check(eqn,ans[1])];
 241 [riccati,0];
 242
 243 /* 22 */
 244 (pn_(22),ans:contrib_ode(eqn:'diff(y,x)-y^2-y*sin(2*x)-cos(2*x),y,x));
 245 [[y=-'diff(%u,x,1)/%u,-'diff(%u,x,1)*sin(2*x)+%u*cos(2*x)+'diff(%u,x,2)=0]];
 246
 247 /* 23 */
 248 assume(a>0,b>0);
 249 [a>0,b>0];
 250 (pn_(23),ans:contrib_ode(eqn:'diff(y,x) + a*y^2 - b,y,x));
 251 [-(log(-((sqrt(a)*sqrt(b)-a*y)/(a*y+sqrt(a)*sqrt(b))))/(2*sqrt(a)*sqrt(b)))=x+%c];
 252 solve(map(exp,2*sqrt(a)*sqrt(b)*ans[1]),y);
 253 [y=sqrt(b)*(%e^(2*sqrt(a)*sqrt(b)*x+2*%c*sqrt(a)*sqrt(b))+1)/(sqrt(a)*(%e^(2*sqrt(a)*sqrt(b)*x+2*%c*sqrt(a)*sqrt(b))-1))];
 254 [method,ode_check(eqn,%[1])];
 255 [separable,0];
 256 forget(a>0,b>0);
 257 [a>0,b>0];
 258
 259 /* 24 */
 260 assume(nu>-2,a>0,b>0);
 261 [nu > -2,a>0,b>0];
 262 (pn_(24),ans:contrib_ode(eqn:'diff(y,x) + a*y^2 - b*x^nu,y,x),0);
 263 0$
 264 /*
 265 [y = -(sqrt(a)*sqrt(b)*(%c*bessel_k((nu+1)/(nu+2),2*sqrt(a)*sqrt(b)*x^(nu/2+1)/(nu+2))-bessel_i(-((nu+1)/(nu+2)),2*sqrt(a)*sqrt(b)*x^(nu/2+1)/(nu+2)))*x^(nu/2)/(%c*a*bessel_k(1/(nu+2),2*sqrt(a)*sqrt(b)*x^(nu/2+1)/(nu+2))+a*bessel_i(1/(nu+2),2*sqrt(a)*sqrt(b)*x^(nu/2+1)/(nu+2))))];
 266  */
 267 [method,ratsimp(ode_check(eqn,ans[1]))];
 268 [riccati,0];
 269 forget(nu>-2,a>0,b>0);
 270 [nu>-2,a>0,b>0];
 271
 272 /* 25 - This is a Riccati equation */
 273 (pn_(25),ans:contrib_ode(eqn:'diff(y,x)+a*y^2-b*x^(2*nu)-c*x^(nu-1),y,x));
 274 [[y='diff(%u,x,1)/(%u*a),%u*a^2*(b*x^(2*nu)+c*x^(nu-1))-'diff(%u,x,2)*a=0]];
 275 method;
 276 riccati;
 277
 278 /* 26 */
 279 (pn_(26),ans:contrib_ode(eqn:'diff(y,x)-(A*y- a)*(B*y-b),y,x));
 280 [((log(B*y-b)-log(A*y-a))/(A*b-a*B)) = x+%c];
 281 [method,ode_check(eqn,ans[1])];
 282 [separable,0];
 283
 284 /* 27 - also Murphy 1.62*/
 285 (pn_(27),ans:contrib_ode(eqn:'diff(y,x) + a*y*(y-x) - 1,y,x),0);
 286 /* [y = (sqrt(%pi)*a*x*%e^(a*x^2/2)*erf(sqrt(a)*x/sqrt(2))
 287          +sqrt(a)*(sqrt(2)*%c*a*x*%e^(a*x^2/2)+sqrt(2)))
 288          /(sqrt(%pi)*a*%e^(a*x^2/2)*erf(sqrt(a)*x/sqrt(2))
 289           +sqrt(2)*%c*a^(3/2)*%e^(a*x^2/2))]; */
 290 0$
 291 [method,ode_check(eqn,ans[1])];
 292 [riccati,0];
 293
 294 /* 28 */
 295 (pn_(28),ans:contrib_ode(eqn:'diff(y,x)+x*y^2-x^3*y-2*x,y,x));
 296 [y = %e^-(x^4/4)*(x^2*%e^(x^4/4)*(sqrt(%pi)*%c*erf(x^2/2)+sqrt(%pi))+2*%c)
 297           /(sqrt(%pi)*%c*erf(x^2/2)+sqrt(%pi))];
 298 [method,ode_check(eqn,ans[1])];
 299 [riccati,0];
 300
 301 /* 29 */
 302 (pn_(29),ans:contrib_ode(eqn:'diff(y,x) - x*y^2 - 3*x*y,y,x));
 303 [-((log(y+3)-log(y))/3) = x^2/2+%c];
 304 solve(map(exp,3*ans[1]),y);
 305 [y = -3*%e^(3*x^2/2+3*%c)/(%e^(3*x^2/2+3*%c)-1)];
 306 [method,ode_check(eqn,%[1])];
 307 [separable,0];
 308
 309 /* 30 */
 310 (pn_(30),ans:contrib_ode(eqn:'diff(y,x)+x^(-a-1)*y^2-x^a,y,x));
 311 [y = -(((%c*bessel_y(a+1,2*sqrt(-x))+bessel_j(a+1,2*sqrt(-x))
 312        -%c*bessel_y(a-1,2*sqrt(-x))-bessel_j(a-1,2*sqrt(-x)))*sqrt(-x)
 313     +a*(%c*bessel_y(a,2*sqrt(-x))+bessel_j(a,2*sqrt(-x))))*x^a
 314     /(2*%c*bessel_y(a,2*sqrt(-x))+2*bessel_j(a,2*sqrt(-x))))];
 315 [method,ode_check(eqn,ans[1])];
 316 [riccati,0];
 317
 318 /* 31 */
 319 assume(n>-1);
 320 [n>-1];
 321 (pn_(31),ans:contrib_ode(eqn:'diff(y,x) - a*x^n*(y^2+1),y,x));
 322 [atan(y)/a = x^(n+1)/(n+1)+%c];
 323 solve(ans[1],y);
 324 [y = tan(a*x^(n+1)/(n+1)+%c*a*n/(n+1)+%c*a/(n+1))];
 325 [method,ode_check(eqn,%[1])];
 326 [separable,0];
 327 forget(n>-1);
 328 [n>-1];
 329
 330 /* 32 - FIXME: Why the %i in the solution? */
 331 assume(cos(x)>0);
 332 [cos(x)>0];
 333 (pn_(32),ans:contrib_ode(eqn:'diff(y,x) + y^2*sin(x) - 2*sin(x)/cos(x)^2,y,x));
 334 [y = ((3*%c-%i)*sinh(3*log(cos(x))/2)+(%c-3*%i)*cosh(3*log(cos(x))/2))/(2*%i*cos(x)*sinh(3*log(cos(x))/2)-2*%c*cos(x)*cosh(3*log(cos(x))/2))];
 335 [method,ode_check(eqn,ans[1])];
 336 [riccati,0];
 337 forget(cos(x)>0);
 338 [cos(x)>0];
 339
 340 /* 33 */
 341 (pn_(33),ans:contrib_ode(eqn:'diff(y,x)-y^2*'diff(f(x),x)/g(x)+'diff(g(x),x)/f(x),y,x));
 342 [[y = -'diff(%u,x,1)*g(x)/(%u*'diff(f(x),x,1)),-'diff(%u,x,1)*('diff(f(x),x,2)/g(x)-'diff(f(x),x,1)*'diff(g(x),x,1)/g(x)^2)-%u*('diff(f(x),x,1))^2*'diff(g(x),x,1)/(f(x)*g(x)^2)+'diff(%u,x,2)*'diff(f(x),x,1)/g(x)=0]];
 343 method;
 344 riccati;
 345
 346 /* 34 */
 347 (pn_(34),ans:contrib_ode(eqn:'diff(y,x)+f(x)*y^2+g(x)*y,y,x));
 348 [y=%e^-('integrate(g(x),x))/('integrate(f(x)*%e^-'integrate(g(x),x),x)+%c)];
 349 [method,ode_check(eqn,ans[1])];
 350 [bernoulli,0];
 351
 352 /* 35 */
 353 assume(4*a^2>4*b);
 354 [4*a^2>4*b];
 355 (pn_(35),ans:contrib_ode(eqn:'diff(y,x)+f(x)*(y^2+2*a*y+b),y,x));
 356 [-(log(-((-y+sqrt(a^2-b)-a)/(y+sqrt(a^2-b)+a)))/(2*sqrt(a^2-b)))='integrate(f(x),x)+%c];
 357 solve(map(exp,2*sqrt(a^2-b)*ans[1]),y);
 358 [y=(sqrt(a^2-b)*(%e^(2*sqrt(a^2-b)*'integrate(f(x),x)+2*%c*sqrt(a^2-b))+1)+a*(1-%e^(2*sqrt(a^2-b)*'integrate(f(x),x)+2*%c*sqrt(a^2-b))))/(%e^(2*sqrt(a^2-b)*'integrate(f(x),x)+2*%c*sqrt(a^2-b))-1)];
 359 [method,ode_check(eqn,%[1])];
 360 [separable,0];
 361 forget(4*a^2>4*b);
 362 [4*a^2>4*b];
 363
 364 /* 36 */
 365 (pn_(36),ans:contrib_ode(eqn:'diff(y,x) + y^3 + a*x*y^2,y,x));
 366 false;
 367
 368 /* 37 */
 369 (pn_(37),ans:contrib_ode(eqn:'diff(y,x)-y^3-a*exp(x)*y^2,y,x));
 370 false;
 371
 372 /* Kamke 38 - Chini invariant constant - 1/(8*a*b^2)
 373
 374    One integrable case is a:-1/6  b:1/3
 375 */
 376 (pn_(38),a:-1/6,b:1/3,ans:contrib_ode(eqn:'diff(y,x) - a*y^3 - b*x^(-3/2),y,x));
 377 [x = %c*%e^((sqrt(x)*(2*y*log(sqrt(x)*y+1)-2*y*log(sqrt(x)*y-2))+2*log(sqrt(x)*y+1)-2*log(sqrt(x)*y-2)-6)/(3*sqrt(x)*y+3))];
 378 method;
 379 genhom;
 380 kill(a,b);
 381 done;
 382
 383 /* 39 */
 384 (pn_(39),ans:contrib_ode(eqn:'diff(y,x)-a3*y^3-a2*y^2-a1*y-a0,y,x));
 385 ['integrate(1/(a3*y^3+a2*y^2+a1*y+a0),y)=x+%c];
 386
 387 /* 40 */
 388 (pn_(40),ans:contrib_ode(eqn:'diff(y,x)+3*a*y^3+6*a*x*y^2,y,x));
 389 false;
 390
 391 /* 41 */
 392 assume(b^2+4*a>0);
 393 [b^2+4*a>0];
 394 (pn_(41),ans:contrib_ode(eqn:'diff(y,x)+a*x*y^3+b*y^2,y,x));
 395 [x=%c*%e^-((b*log(-((-2*a*x*y+sqrt(b^2+4*a)-b)/(2*a*x*y+sqrt(b^2+4*a)+b)))+sqrt(b^2+4*a)*(log(a*x^2*y^2+b*x*y-1)-2*log(x*y)))/(2*sqrt(b^2+4*a)))];
 396 method;
 397 genhom;
 398 forget(b^2+4*a>0);
 399 [b^2+4*a>0];
 400
 401 /* 42 */
 402 (pn_(42),ans:contrib_ode(eqn:'diff(y,x)-x*(x+2)*y^3-(x+3)*y^2,y,x));
 403 false;
 404
 405 /* 43 */
 406 (pn_(43),ans:contrib_ode(eqn:'diff(y,x)+(3*a*x^2+4*a^2*x+b)*y^3+3*x*y^2,y,x));
 407 false;
 408
 409 /* 44 */
 410 (pn_(44),ans:contrib_ode(eqn:'diff(y,x)+2*a*x^3*y^3+2*x*y,y,x));
 411 [y=%e^-x^2/sqrt(%c-a*(2*x^2+1)*%e^-(2*x^2)/2)];
 412 [method,ode_check(eqn,ans[1])];
 413 [bernoulli,0];
 414
 415 /* 45 */
 416 (pn_(45),ans:contrib_ode(eqn:'diff(y,x)+2*(a^2*x^3-b^2*x)*y^3+3*b*y^2,y,x));
 417 false;
 418
 419 /* 46 Abel eqn with relative invariant = 0 */
 420 (kn_("46i"),a:1,ans:contrib_ode(eqn:'diff(y,x)- x^a*y^3+3*y^2-x^(-a)*y-x^(-2*a)+ a*x^(-a-1),y,x));
 421 [x = %c*%e^-((2*log(x*y-1)-log(x*y-2)-log(x*y))/2)];
 422 /* This doesn't work.  Bug is ode_check??  ***FIXME***
 423 [method,ode_check(eqn,ans[1])];
 424 [genhom,0]; */
 425 radcan(ans[1])^2;
 426 x^2 = %c^2*x*y*(x*y-2)/(x*y-1)^2;
 427 ans:solve(%,y);
 428 [y = (%c*sqrt(%c^2-x^2)+x^2-%c^2)/(x^3-%c^2*x), y = -((%c*sqrt(%c^2-x^2)-x^2+%c^2)/(x^3-%c^2*x))];
 429 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 430 [genhom,0,0];
 431 kill(a);
 432 done;
 433
 434 (pn_("46ii"),assume(a>1),ans:contrib_ode(eqn:'diff(y,x) - x^a*y^3 + 3*y^2 - x^(-a)*y -x^(-2*a) + a*x^(-a-1),y,x));
 435 [y=%e^-(2*x^(1-a)/(1-a))/sqrt(2*(a-1)^((a+1)/(1-a))
 436   *gamma_incomplete((a+1)/(1-a),-4*x^(1-a)/(a-1))*x^(a+1)
 437    /((1-a)*4^((a+1)/(1-a))*(-x^(1-a))^((a+1)/(1-a)))+%c)+1/x^a];
 438 [method,ode_check(eqn,ans[1])];
 439 [abel,0];
 440 forget(a>1);
 441 [a>1];
 442
 443 /* 47 - Too general - E S Cheb-Terrab and T Kolokolnikov */
 444 /* 48 - Too general - E S Cheb-Terrab and T Kolokolnikov */
 445 /* 49 - Crash */
 446 /* 50 - Too general - E S Cheb-Terrab and T Kolokolnikov */
 447 /* 51 infinite run time */
 448
 449 /* 52 Asks if n is integer */
 450 /* (pn_(52),ans:'diff(y,x)-a*y^n-b*x^(n/(1-n)),y,x)); */
 451
 452 /* 53 */
 453 /* (pn_(53),ans:contrib_ode(eqn:'diff(y,x)-f(x)^(1-n)*'diff(g(x),x)*y^n/(a*g(x)+b)^n-'diff(f(x),x)*y/f(x)-f(x)*'diff(g(x),x),y,x));
 454 false; */
 455
 456 /* 54 */
 457 /* (pn_(54),ans:contrib_ode(eqn:'diff(y,x)-a^n*f(x)^(1-n)*'diff(g(x),x)*y^n-'diff(f(x),x)*y/f(x)-f(x)*'diff(g(x),x),y,x));
 458 false; */
 459
 460 /* 55 - Too general - E S Cheb-Terrab and T Kolokolnikov */
 461 /* 56 - Too general - E S Cheb-Terrab and T Kolokolnikov */
 462
 463 /* 57 */
 464 (pn_(57),ans:contrib_ode(eqn:'diff(y,x)-sqrt(abs(y)),y,x));
 465 ['integrate(1/sqrt(abs(y)),y)=x+%c];
 466 [method,ode_check(eqn,ans[1])];
 467 [separable,0];
 468
 469 /* 58 */
 470 assume((8*b+a^2)>0,x>0);
 471 [(8*b+a^2)>0,x>0];
 472 (pn_(58),ans:contrib_ode(eqn:'diff(y,x)-a*sqrt(y)-b*x,y,x));
 473 [x=%c*%e^-(((sqrt(8*b+a^2)*log(-((-2*y+a*x*sqrt(y)+b*x^2)/x^2))+a*log(-((-4*sqrt(y)+sqrt(8*b+a^2)*x+a*x)/(4*sqrt(y)+sqrt(8*b+a^2)*x-a*x))))/(2*sqrt(8*b+a^2))))];
 474 forget((8*b+a^2)>0,x>0);
 475 [(8*b+a^2)>0,x>0];
 476
 477 /* 59 */
 478 (pn_(59),ans:contrib_ode(eqn:'diff(y,x)-a*sqrt(y^2+1)-b,y,x));
 479 ['integrate(1/(a*sqrt(y^2+1)+b),y) = x+%c];
 480 /* maxima no longer integrates this 2006-09-02
 481 assume(y>0);
 482 [y>0];
 483 ans[1],abs,integrate;
 484 2*sqrt(y)=x+%c;
 485 [method,ode_check(eqn,%)];
 486 [separable,0];
 487 forget(y>0);
 488 [y>0];*/
 489
 490 /* 60 */
 491 (pn_(60),ans:contrib_ode(eqn:'diff(y,x)-sqrt(y^2-1)/sqrt(x^2-1),y,x));
 492 [log(2*sqrt(y^2-1)+2*y)=log(2*sqrt(x^2-1)+2*x)+%c];
 493 [method,ode_check(eqn,ans[1])];
 494 [separable,0];
 495
 496 /* 61 */
 497 (pn_(61),ans:contrib_ode(eqn:'diff(y,x)-sqrt(x^2-1)/sqrt(y^2-1),y,x));
 498 [(y*sqrt(y^2-1)-log(2*sqrt(y^2-1)+2*y))/2=(x*sqrt(x^2-1)-log(2*sqrt(x^2-1)+2*x))/2+%c];
 499 [method,ode_check(eqn,ans[1])];
 500 [separable,0];
 501
 502 /* 62 "infinite" run time */
 503
 504 /* 63 */
 505 (pn_(63),ans:contrib_ode(eqn:'diff(y,x)-(1+ y^2)/(abs(y+sqrt(1+y))*sqrt(1+x)^3),y,x));
 506 ['integrate(abs(sqrt(y+1)+y)/(y^2+1),y)= %c-2/sqrt(x+1)];
 507 [method,ode_check(eqn,%)];
 508 [separable,0];
 509
 510 /* 64 */
 511 /* Used to pass but now fails due to integration problems.
 512    Almost certainly related to bug # 1487703
 513
 514 assume(a>0,b>0,4*a*c>b^2);
 515 [a>0,b>0,4*a*c>b^2];
 516 (pn_(64),ans:contrib_ode(eqn: 'diff(y,x)-sqrt((a*y^2+b*y+c)/(a*x^2+b*x+c)),y,x));
 517 [(sqrt(a)*sqrt(2*a*y^2+2*b*y+2*c)*asinh((4*a*y+2*b)/sqrt(16*a*c-4*b^2))-sqrt(2)*sqrt(a)*asinh((2*a*x+b)/sqrt(4*a*c-b^2))*sqrt(a*y^2+b*y+c))/(sqrt(2)*a*sqrt(2*a*y^2+2*b*y+2*c))=%c];
 518 method;
 519 exact;
 520 forget(a>0,b>0,4*a*c>b^2);
 521 [a>0,b>0,4*a*c>b^2];
 522 */
 523
 524 /* 65 */
 525 (pn_(65),ans:contrib_ode(eqn:'diff(y,x)-sqrt((y^3+1)/(x^3+1)),y,x));
 526 [-((sqrt(2*y^3+2)*'integrate((3*y^2*'integrate(sqrt((y^3+1)/(x^3+1)),x)+(-3*y^5-3*y^2)*'integrate(1/((x^3+1)*sqrt((y^3+1)/(x^3+1))),x)-2*y^3-2)/(2*y^3+2)^(3/2),y)+'integrate(sqrt((y^3+1)/(x^3+1)),x))/sqrt(2*y^3+2))=%c];
 527 method;
 528 exact;
 529
 530 /* 66 */
 531 (pn_(66),ans:contrib_ode(eqn:'diff(y,x)-sqrt(abs(y*(1-y)*(1-a*y)))/sqrt(abs(x*(1-x)*(1-a*x))),y,x));
 532 ['integrate(1/(sqrt(abs(y-1))*sqrt(abs(y))*sqrt(abs(a*y-1))),y)='integrate(1/(sqrt(abs(x-1))*sqrt(abs(x))*sqrt(abs(a*x-1))),x)+%c];
 533 [method,ode_check(eqn,%)];
 534 [separable,0];
 535
 536 /* 67 */
 537 (pn_(67),ans:contrib_ode(eqn:'diff(y,x)-sqrt(1-y^4)/sqrt(1-x^4),y,x));
 538 ['integrate(1/sqrt(1-y^4),y) = 'integrate(1/sqrt(1-x^4),x)+%c];
 539 [method,ode_check(eqn,ans[1])];
 540 [separable,0];
 541
 542 /* 68 */
 543 /*(pn_(68),ans:contrib_ode(eqn:'diff(y,x)-sqrt((a*y^4+b*y^2+1)/(a*x^4+b*x^2+1)),y,x));
 544 ['integrate(1/sqrt(a*y^4+b*y^2+1),y)='integrate(1/sqrt(a*x^4+b*x^2+1),x)+%c]; */
 545
 546 /* 69 */
 547
 548 /* 74 - Too general - E S Cheb-Terrab and T Kolokolnikov */
 549
 550 /* 75 */
 551 (pn_(75),ans:contrib_ode(eqn:'diff(y,x)-exp(x-y)+exp(x),y,x));
 552 [log(%e^y-1)+%e^x=%c];
 553 solve(ans[1],y);
 554 [y=log(%e^(%c-%e^x)+1)];
 555 [method,ode_check(eqn,%[1])];
 556 [exact,0];
 557
 558 /* 76 */
 559 assume(4*a^2>4*b^2);
 560 [4*a^2>4*b^2];
 561 (pn_(76),ans:contrib_ode(eqn:'diff(y,x)-a*cos(y)+b,y,x));
 562 [-(log(-(((-b-a)*sin(y)+sqrt(a^2-b^2)*(cos(y)+1))/((b+a)*sin(y)+sqrt(a^2-b^2)*(cos(y)+1))))/sqrt(a^2-b^2))=x+%c];
 563 method;
 564 separable;
 565 forget(4*a^2>4*b^2);
 566 [4*a^2>4*b^2];
 567
 568 /* 77 infinite run time */
 569 /* 78 infinite run time */
 570 /* 79 - Too general - E S Cheb-Terrab and T Kolokolnikov */
 571
 572 /* 80 */
 573 (pn_(80),ans:contrib_ode(eqn:'diff(y,x)+f(x)*sin(y)+(1-'diff(f(x),x))*cos(y)-'diff(f(x),x)-1,y,x));
 574 false;
 575
 576 /* 81 - Crash */
 577 /* 82 - Too general - E S Cheb-Terrab and T Kolokolnikov */
 578
 579 /* 83 */
 580 (pn_(83),ans:contrib_ode(eqn:'diff(y,x)-tan(x*y),y,x));
 581 false;
 582
 583 /* 84 */
 584 use_pdiff:true;
 585 true;
 586 (pn_(84),ans:contrib_ode(eqn:'diff(y,x)-f(a*x+b*y),y,x));
 587 [b*'integrate(f(b*y+a*x)/(b*f(b*y+a*x)+a),x)-'integrate(((a*b^3*f(b*y+a*x)+a^2*b^2)*'integrate(pderivop(f,1)(b*y+a*x)/(b^2*f(b*y+a*x)^2+2*a*b*f(b*y+a*x)+a^2),x)+b)/(b*f(b*y+a*x)+a),y)=%c];
 588 method;
 589 lie;
 590 use_pdiff:false;
 591 false;
 592
 593 /* 85 - Crash */
 594 /* 86 infinite run time */
 595 /* 87 infinite run time */
 596
 597 /* 88 */
 598 (pn_(88),ans:contrib_ode(eqn:2*'diff(y,x)-3*y^2-4*a*y-b-c*exp(-2*a*x),y,x));
 599 [[y=-2*'diff(%u,x,1)/(3*%u),9*%u*(c*%e^-(2*a*x)/2+b/2)/4-3*'diff(%u,x,1)*a+3*'diff(%u,x,2)/2=0]];
 600 method;
 601 riccati;
 602
 603 /* 89 */
 604 assume(a>0);
 605 [a>0];
 606 (pn_(89),ans:contrib_ode(eqn: x*'diff(y,x)-sqrt(a^2-x^2),y,x));
 607 [y = -a*log(2*a*sqrt(a^2-x^2)/abs(x)+2*a^2/abs(x))+sqrt(a^2-x^2)+%c];
 608 [method,ode_check(eqn,%[1])];
 609 [linear,0];
 610 forget(a>0);
 611 [a>0];
 612
 613 /* 90 */
 614 (pn_(90),ans:contrib_ode(eqn:x*'diff(y,x)+y-x*sin(x),y,x));
 615 [y=(sin(x)-x*cos(x)+%c)/x];
 616 [method,ode_check(eqn,ans[1])];
 617 [linear,0];
 618
 619 /* 91 */
 620 (pn_(91),ans:contrib_ode(eqn:x*'diff(y,x)-y-x/log(x),y,x));
 621 [y=x*(log(log(x))+%c)];
 622 [method,ode_check(eqn,ans[1])];
 623 [linear,0];
 624
 625 /* 92 */
 626 (pn_(92),ans:contrib_ode(eqn:x*'diff(y,x)-y-x^2*sin(x),y,x));
 627 [y=x*(%c-cos(x))];
 628 [method,ode_check(eqn,ans[1])];
 629 [linear,0];
 630
 631 /* 93 */
 632 (pn_(93),ans:contrib_ode(eqn:x*'diff(y,x)-y-x*cos(log(log(x)))/log(x),y,x));
 633 [y=x*(sin(log(log(x)))+%c)];
 634 [method,ode_check(eqn,ans[1])];
 635 [linear,0];
 636
 637 /* 94 */
 638 (pn_(94),ans:contrib_ode(eqn:x*'diff(y,x)+a*y+b*x^n,y,x));
 639 [y=%e^-(a*log(x))*(%c-b*%e^(n*log(x)+a*log(x))/(n+a))];
 640 [method,radcan(ode_check(eqn,ans[1]))];
 641 [linear,0];
 642
 643 /* 95 */
 644 (pn_(95),ans:contrib_ode(eqn:x*'diff(y,x)+y^2+x^2,y,x));
 645 [y=-((bessel_y(1,x)*%c+bessel_j(1,x))*x/(bessel_y(0,x)*%c+bessel_j(0,x)))];
 646 [method,ode_check(eqn,ans[1])];
 647 [riccati,0];
 648
 649 /* 96 */
 650 (pn_(96),ans:contrib_ode(eqn:x*'diff(y,x)-y^2+1,y,x));
 651 [-((log(y+1)-log(y-1))/2)=log(x)+%c];
 652 solve(map(exp,-2*ans[1]),y);
 653 [y=-((%e^(2*%c)*x^2+1)/(%e^(2*%c)*x^2-1))];
 654 [method,ode_check(eqn,%[1])];
 655 [separable,0];
 656
 657 /* 97 */
 658 assume(a>0,b>0);
 659 [a>0,b>0];
 660 (pn_(97),ans:contrib_ode(eqn:x*'diff(y,x)+a*y^2-y+b*x^2,y,x));
 661 [y=-sqrt(b)*x*tan(sqrt(a)*sqrt(b)*x-%c)/sqrt(a)];
 662 [method,ode_check(eqn,ans[1])];
 663 [riccati,0];
 664 forget(a>0,b>0);
 665 [a>0,b>0];
 666
 667 /* 98 */
 668 assume(a>0,b>0,c>0);
 669 [a>0,b>0,c>0];
 670 (pn_(98),ans:contrib_ode(eqn:x*'diff(y,x)+a*y^2-b*y+c*x^(2*b),y,x));
 671 [y=-sqrt(c)*x^b*tan(sqrt(a)*sqrt(c)*x^b/b-%c)/sqrt(a)];
 672 [method,ode_check(eqn,ans[1])];
 673 [riccati,0];
 674 forget(a>0,b>0,c>0);
 675 [a>0,b>0,c>0];
 676
 677 /* 99  Requires specific values of beta and b */
 678 (beta:4,b:3,assume(a>0,c>0),done);
 679 done;
 680 (pn_(99),ans:contrib_ode(eqn:x*'diff(y,x)+a*y^2-b*y-c*x^beta,y,x));
 681 [y = -((((bessel_k(7/4,sqrt(a)*sqrt(c)*x^2/2)+bessel_k(1/4,sqrt(a)*sqrt(c)*x^2/2))*%c
 682   -bessel_i(7/4,sqrt(a)*sqrt(c)*x^2/2)-bessel_i(-(1/4),sqrt(a)*sqrt(c)*x^2/2))*sqrt(a)*sqrt(c)*x^2
 683  -3*bessel_k(3/4,sqrt(a)*sqrt(c)*x^2/2)*%c-3*bessel_i(3/4,sqrt(a)*sqrt(c)*x^2/2))
 684  /((2*'bessel_k(3/4,sqrt(a)*sqrt(c)*x^2/2)*%c+2*'bessel_i(3/4,sqrt(a)*sqrt(c)*x^2/2))*a))];
 685 [method,ode_check(eqn,ans[1])];
 686 [riccati,0];
 687 (kill(beta,b),forget(a>0,c>0),done);
 688 done;
 689
 690 /* 100 */
 691 assume(a>0);
 692 [a>0];
 693 (pn_(100),ans:contrib_ode(eqn:x*'diff(y,x)+x*y^2+a,y,x));
 694 [y = -((((bessel_y(2,2*sqrt(a)*sqrt(x))-bessel_y(0,2*sqrt(a)*sqrt(x)))*%c
 695         +bessel_j(2,2*sqrt(a)*sqrt(x))-bessel_j(0,2*sqrt(a)*sqrt(x)))*sqrt(a)*sqrt(x)
 696         -bessel_y(1,2*sqrt(a)*sqrt(x))*%c-bessel_j(1,2*sqrt(a)*sqrt(x)))
 697       /((2*bessel_y(1,2*sqrt(a)*sqrt(x))*%c+2*bessel_j(1,2*sqrt(a)*sqrt(x)))*x))];
 698 [method,ode_check(eqn,ans[1])];
 699 [riccati,0];
 700 forget(a>0);
 701 [a>0];