share/contrib/diffequations/tests/rtestode_murphy_1_1.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Murphy equations 1.1 - 1.100
   5
   6   Reference:
   7     G M Murphy, Ordinary Differential Equations and Their
   8     Solutions, Van Nostrand, 1960
   9
  10   First Order and of First Degree, p 224 ff
  11
  12 */
  13 /* Don't kill(all); It messes up trigsimp */
  14
  15 /* Print ode number*/
  16 (pn_(n_):=print("Murphy ODE 1.",n_),true);
  17 true;
  18
  19 /*   1 */
  20 (pn_(1),ans:contrib_ode(eqn:'diff(y,x)=a*f(x),y,x));
  21  [y=a*'integrate(f(x),x)+%c];
  22 [method,ode_check(eqn,ans[1])];
  23 [linear,0];
  24
  25 /*   2 */
  26 (pn_(2),ans:contrib_ode(eqn:'diff(y,x)=x+sin(x)+y,y,x));
  27 [y = %e^x*(%e^-x*(-sin(x)-cos(x))/2+(-x-1)*%e^-x+%c)];
  28 [method,ode_check(eqn,ans[1])];
  29 [linear,0];
  30
  31 /*   3 */
  32 (pn_(3),ans:contrib_ode(eqn:'diff(y,x)=x^2+3*cosh(x)+2*y,y,x),0);
  33 0$
  34 ans:expand(ans[1]);
  35 y = %c*%e^(2*x)-3*%e^x/2-%e^-x/2-x^2/2-x/2-1/4;
  36 [method,ode_check(eqn,ans)];
  37 [linear,0];
  38
  39 /* 4 */
  40 (pn_(4),ans:contrib_ode(eqn:'diff(y,x)=a+b*x+c*y,y,x));
  41 [y=(-b*(c*x+1)*%e^-(c*x)/c^2-a*%e^-(c*x)/c+%c)*%e^(c*x)];
  42 [method,ode_check(eqn,ans[1])];
  43 [linear,0];
  44
  45 /* 5 */
  46 (pn_(5),ans:contrib_ode(eqn:'diff(y,x)=a*cos(b*x+c)+k*y,y,x));
  47 [y=%e^(k*x)*(a*%e^-(k*x)*(b*sin(b*x+c)-k*cos(b*x+c))/(k^2+b^2)+%c)];
  48 [method,ode_check(eqn,ans[1])];
  49 [linear,0];
  50
  51 /* 6 */
  52 (pn_(6),ans:contrib_ode(eqn:'diff(y,x)=a*sin(b*x+c)+k*y,y,x));
  53 [y=%e^(k*x)*(a*%e^-(k*x)*(-k*sin(b*x+c)-b*cos(b*x+c))/(k^2+b^2)+%c)];
  54 [method,ode_check(eqn,ans[1])];
  55 [linear,0];
  56
  57 /* 7 */
  58 (pn_(7),assume(c>0,k>c),ans:contrib_ode(eqn:'diff(y,x)=a+b*exp(k*x)+c*y,y,x),forget(c>0,k>c),0);
  59 0$
  60 expand(ans[1]);
  61 y = (-(b*%e^(k*x))/(c-k))+%c*%e^(c*x)-a/c$
  62 [method,ode_check(eqn,ans[1])];
  63 [linear,0];
  64
  65 /* 8 */
  66 (pn_(8),ans:contrib_ode(eqn:'diff(y,x)=x*(x^2-2*y),y,x));
  67 [y = %e^-x^2*((x^2-1)*%e^x^2/2+%c)];
  68 [method,ode_check(eqn,ans[1])];
  69 [linear,0];
  70
  71 /* 9 NOTE: Case a=2 is special */
  72 (pn_(9),ans:contrib_ode(eqn:'diff(y,x)=x*(exp(-x^2)-a*y),y,x));
  73 [y = %e^-(a*x^2/2)*(%e^(a*x^2/2-x^2)/(a-2)+%c)];
  74 [method,ode_check(eqn,ans[1])];
  75 [linear,0];
  76
  77 ans:contrib_ode(eqn:'diff(y,x)=x*(exp(-x^2)-2*y),y,x);
  78 [y=(x^2/2+%c)*%e^-x^2];
  79 [method,ode_check(eqn,ans[1])];
  80 [linear,0];
  81
  82 /* 10 */
  83 (pn_(10),ans:contrib_ode(eqn:'diff(y,x)=x^2*(a*x^3+b*y),y,x));
  84 [y = (%c-a*(3*b*x^3+9)*%e^-(b*x^3/3)/(3*b^2))*%e^(b*x^3/3)];
  85 [method,ode_check(eqn,ans[1])];
  86 [linear,0];
  87
  88 /* 11(i)  n#-1*/
  89 (pn_(11),assume(not equal(n,-1)),ans:contrib_ode(eqn:'diff(y,x)=a*x^n*y,y,x),forget(not equal(n,-1)),ans);
  90 [y = %c*%e^(a*x^(n+1)/(n+1))];
  91 [method,ode_check(eqn,ans[1])];
  92 [linear,0];
  93
  94 /* 11(ii) n=-1*/
  95 ans:contrib_ode(eqn:'diff(y,x)=a*x^-1*y,y,x);
  96 [y = %c*%e^(a*log(x))];
  97 [method,ode_check(eqn,ans[1])];
  98 [linear,0];
  99
 100 /* 12 */
 101 (pn_(12),ans:contrib_ode(eqn:'diff(y,x)=cos(x)*sin(x)+y*cos(x),y,x));
 102 [y = (%e^-sin(x)*(-sin(x)-1)+%c)*%e^sin(x)];
 103 ans:expand(ans[1]);
 104 y = %c*%e^sin(x)-sin(x)-1;
 105 [method,ode_check(eqn,ans)];
 106 [linear,0];
 107
 108 /* 13 */
 109 (pn_(13),ans:contrib_ode(eqn:'diff(y,x)=exp(sin(x))+y*cos(x),y,x));
 110 [y = (x+%c)*%e^sin(x)];
 111 [method,ode_check(eqn,ans[1])];
 112 [linear,0];
 113
 114 /* 14 */
 115 (pn_(14),ans:contrib_ode(eqn:'diff(y,x)=y*cot(x),y,x));
 116 [y = %c*sin(x)];
 117 [method,ode_check(eqn,ans[1])];
 118 [linear,0];
 119
 120 /* 15 */
 121 (pn_(15),ans:contrib_ode(eqn:'diff(y,x)=1-y*cot(x),y,x));
 122 [y = (%c-cos(x))/sin(x)];
 123 [method,ode_check(eqn,ans[1])];
 124 [linear,0];
 125
 126 /* 16 */
 127 (pn_(16),ans:contrib_ode(eqn:'diff(y,x)=x*csc(x)-y*cot(x),y,x));
 128 [y = (x^2/2+%c)/sin(x)];
 129 [method,ode_check(eqn,ans[1])];
 130 [linear,0];
 131
 132 /* 17 */
 133 (pn_(17),ans:contrib_ode(eqn:'diff(y,x)=(2*csc(2*x)+cot(x))*y,y,x));
 134 [y = %c*sin(x)/(csc(2*x)+cot(2*x))];
 135 [method,ode_check(eqn,ans[1])];
 136 [linear,0];
 137
 138 /* 18 */
 139 (pn_(18),ans:contrib_ode(eqn:'diff(y,x)=sec(x)-y*cot(x),y,x));
 140 [y = (%c-log(1-sin(x)^2)/2)/sin(x)];
 141 ans:subst(cos(x)^2,1-sin(x)^2,ans[1]);
 142 y = (%c-log(cos(x)))/sin(x);
 143 [method,ode_check(eqn,ans)];
 144 [linear,0];
 145
 146 /* 19 */
 147 (pn_(19),ans:contrib_ode(eqn:'diff(y,x)=exp(x)*sin(x)+y*cot(x),y,x));
 148 [y = (%e^x+%c)*sin(x)];
 149 [method,ode_check(eqn,ans[1])];
 150 [linear,0];
 151
 152 /* 20 */
 153 (pn_(20),ans:contrib_ode(eqn:'diff(y,x)+csc(x)+2*y*cot(x)=0,y,x));
 154 [y = (cos(x)+%c)/sin(x)^2];
 155 [method,ode_check(eqn,ans[1])];
 156 [linear,0];
 157
 158 /* 21 */
 159 (pn_(21),ans:contrib_ode(eqn:'diff(y,x)=2*csc(2*x)*sec(x)^2-2*y*cot(2*x),y,x));
 160 [y=(4*sin(2*x)/(sin(2*x)^2+cos(2*x)^2+2*cos(2*x)+1)+%c)/sin(2*x)];
 161 [method,ode_check(eqn,ans[1])];
 162 [linear,0];
 163
 164 /* 22 */
 165 (pn_(22),ans:contrib_ode(eqn:'diff(y,x)=2*(cot(x)^2*cos(2*x)-y*csc(2*x)),y,x));
 166 [y = (csc(2*x)+cot(2*x))*(log(sin(x)^2+cos(x)^2+2*cos(x)+1)
 167        +log(sin(x)^2+cos(x)^2-2*cos(x)+1)+cos(2*x)+%c)];
 168 [method,ode_check(eqn,ans[1])];
 169 [linear,0];
 170
 171 /* 23 */
 172 (pn_(23),ans:contrib_ode(eqn:'diff(y,x)=2*csc(2*x)*(sin(x)^3+y),y,x),ans:map(lambda([e],map(trigrat,e)),ans));
 173 [y=-((sin(2*x)-2*%c*sin(x))/(2*cos(x)))];
 174 [method,ode_check(eqn,ans[1])];
 175 [linear,0];
 176
 177 /* 24 */
 178 (pn_(24),ans:contrib_ode(eqn:'diff(y,x)=2*csc(2*x)*(1-tan(x)^2+y),y,x),ans:map(lambda([e],map(trigrat,e)),ans));
 179 [y=(%c*sin(2*x)-2)/(cos(2*x)+1)];
 180 [method,ode_check(eqn,ans[1])];
 181 [linear,0];
 182
 183 /* 25 */
 184 (pn_(25),ans:contrib_ode(eqn:'diff(y,x)=y*sec(x),y,x));
 185 [y = %c*(tan(x)+sec(x))];
 186 [method,ode_check(eqn,ans[1])];
 187 [linear,0];
 188
 189 /* 26 */
 190 (pn_(26),ans:contrib_ode(eqn:'diff(y,x)+tan(x)=(1-y)*sec(x),y,x));
 191 [y=(x+%c)/(tan(x)+sec(x))];
 192 [method,ode_check(eqn,ans[1])];
 193 [linear,0];
 194
 195 /* 27 */
 196 (pn_(27),ans:contrib_ode(eqn:'diff(y,x)=y*tan(x),y,x));
 197 [y=%c*sec(x)];
 198 [method,ode_check(eqn,ans[1])];
 199 [linear,0];
 200
 201 /* 28 */
 202 (pn_(28),ans:contrib_ode(eqn:'diff(y,x)=cos(x)+y*tan(x),y,x));
 203 [y=sec(x)*((sin(2*x)/2+x)/2+%c)];
 204 [method,ode_check(eqn,ans[1])];
 205 [linear,0];
 206
 207 /* 29 */
 208 (pn_(29),ans:contrib_ode(eqn:'diff(y,x)=cos(x)-y*tan(x),y,x));
 209 [y=(x+%c)/sec(x)];
 210 [method,ode_check(eqn,ans[1])];
 211 [linear,0];
 212
 213 /* 30 */
 214 (pn_(30),ans:contrib_ode(eqn:'diff(y,x)=sec(x)-y*tan(x),y,x));
 215 [y=(tan(x)+%c)/sec(x)];
 216 [method,ode_check(eqn,ans[1])];
 217 [linear,0];
 218
 219 /* 31 */
 220 (pn_(31),ans:contrib_ode(eqn:'diff(y,x)=sin(2*x)+y*tan(x),y,x));
 221 [y=(%c-2*cos(x)^3/3)*sec(x)];
 222 [method,ode_check(eqn,ans[1])];
 223 [linear,0];
 224
 225 /* 32 */
 226 (pn_(32),ans:contrib_ode(eqn:'diff(y,x)=sin(2*x)-y*tan(x),y,x));
 227 [y=(%c-2*cos(x))/sec(x)];
 228 [method,ode_check(eqn,ans[1])];
 229 [linear,0];
 230
 231 /* 33 */
 232 (pn_(33),ans:contrib_ode(eqn:'diff(y,x)=sin(x)+2*y*tan(x),y,x));
 233 [y=(%c-cos(x)^3/3)*sec(x)^2];
 234 [method,ode_check(eqn,ans[1])];
 235 [linear,0];
 236
 237 /* 34 */
 238 (pn_(34),ans:contrib_ode(eqn:'diff(y,x)=2*(1+sec(2*x)+y*tan(2*x)),y,x));
 239 [y=sec(2*x)*(sin(2*x)+2*x+%c)];
 240 [method,ode_check(eqn,ans[1])];
 241 [linear,0];
 242
 243 /* 35 */
 244 (pn_(35),ans:contrib_ode(eqn:'diff(y,x)=csc(x)+3*y*tan(x),y,x));
 245 [y=sec(x)^3*(log(sin(x))-sin(x)^2/2+%c)];
 246 [method,ode_check(eqn,ans[1])];
 247 [linear,0];
 248
 249 /* 36 */
 250 (pn_(36),ans:contrib_ode(eqn:'diff(y,x)=(a+cos(log(x))+sin(log(x)))*y,y,x));
 251 [y=%c*%e^(x*(sin(log(x))+cos(log(x)))/2+x*(sin(log(x))-cos(log(x)))/2+a*x)];
 252 (assume(x>0),ans:map(trigrat,ans[1]),forget(x>0),ans);
 253 y=%c*%e^(x*sin(log(x))+a*x);
 254 [method,ode_check(eqn,ans)];
 255 [linear,0];
 256
 257 /* 37 */
 258 (pn_(37),ans:contrib_ode(eqn:'diff(y,x)=6*exp(2*x)-y*tanh(x),y,x),0);
 259 0$
 260 expand(ans[1]);
 261 y=%e^(3*x)/cosh(x)+(3*%e^x)/cosh(x)+%c/cosh(x)$
 262 [method,ode_check(eqn,ans[1])];
 263 [linear,0];
 264
 265 /* 38 - too general*/
 266 /* 39 - too general */
 267
 268 /* 41  Riccati */
 269
 270 /* 42  Riccati */
 271 (pn_(42),ans:contrib_ode(eqn:'diff(y,x)+1-x=(x+y)*y,y,x),0);
 272 0$
 273 [method,ode_check(eqn,ans[1])];
 274 [riccati,0];
 275
 276 /* 43  Riccati */
 277 (pn_(43),ans:contrib_ode(eqn:'diff(y,x)=(x+y)^2,y,x));
 278 [[x=%c-atan(sqrt(%t)),y=-x-sqrt(%t)],[x=atan(sqrt(%t))+%c,y=sqrt(%t)-x]];
 279 solve(ans[1][1],sqrt(%t));
 280 [sqrt(%t)=-tan(x-%c)];
 281 ans[1]:ev(ans[1][2],%[1]);
 282 y=tan(x-%c)-x;
 283 solve(ans[2][1],sqrt(%t));
 284 [sqrt(%t)=tan(x-%c)];
 285 ans[2]:ev(ans[2][2],%[1]);
 286 y=tan(x-%c)-x;
 287 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 288 [lagrange,0,0];
 289
 290 /* 44  Riccati */
 291 (pn_(44),ans:contrib_ode(eqn:'diff(y,x)=(x-y)^2,y,x));
 292 [[x = log(sqrt(%t)+1)/2-log(sqrt(%t)-1)/2+%c,y = x-sqrt(%t)],[x=-log(sqrt(%t)+1)/2+log(sqrt(%t)-1)/2+%c,y=x+sqrt(%t)]];
 293 solve(ans[1][2],sqrt(%t));
 294 [sqrt(%t) = x-y];
 295 ans[1][1],%[1];
 296 x=log(-y+x+1)/2-log(-y+x-1)/2+%c;
 297 expand(2*(%-%c));
 298 2*x-2*%c = log(-y+x+1)-log(-y+x-1);
 299 map(exp,%);
 300 %e^(2*x-2*%c) = (-y+x+1)/(-y+x-1);
 301 ans[1]:first(solve(%,y));
 302 y=((x-1)*%e^(2*x-2*%c)-x-1)/(%e^(2*x-2*%c)-1);
 303 solve(ans[2][2],sqrt(%t));
 304 [sqrt(%t) = y-x];
 305 ans[2][1],%[1];
 306 x=-log(y-x+1)/2+log(y-x-1)/2+%c;
 307 expand(2*(%-%c));
 308 2*x-2*%c = log(y-x-1)-log(y-x+1);
 309 map(exp,%);
 310 %e^(2*x-2*%c)=(y-x-1)/(y-x+1);
 311 ans[2]:first(solve(%,y));
 312 y=((x-1)*%e^(2*x-2*%c)-x-1)/(%e^(2*x-2*%c)-1);
 313 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 314 [lagrange,0,0];
 315
 316 /* 45 */
 317 (pn_(45),ans:contrib_ode(eqn:'diff(y,x)=3*(1-x+y)+(x-y)^2,y,x));
 318 [y = (x*(%c*%e^x+1)-2*%c*%e^x-1)/(%c*%e^x+1) ];
 319 [method,ode_check(eqn,ans[1])];
 320 [riccati,0];
 321
 322 /* 46 riccati */
 323 (pn_(46),ans:contrib_ode(eqn:'diff(y,x)=2*x-(1+x^2)*y+y^2,y,x));
 324 [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)];
 325 [method,ode_check(eqn,ans[1])];
 326 [riccati,0];
 327
 328 /* 47 Riccati */
 329 (pn_(47),ans:contrib_ode(eqn:'diff(y,x)=x*(2+x^3)-(2*x^2-y)*y,y,x));
 330 [y=(%c*x^3+x^2-%c)/(%c*x+1)];
 331 [method,ode_check(eqn,ans[1])];
 332 [riccati,0];
 333
 334 /* 48 Riccati */
 335 (pn_(48),ans:contrib_ode(eqn:'diff(y,x)=1+x*(2-x^3)+(2*x^2-y)*y,y,x));
 336 [y = (x^2*(%c*%e^(2*x)+1)+%c*%e^(2*x)-1)/(%c*%e^(2*x)+1) ];
 337 [method,ode_check(eqn,ans[1])];
 338 [riccati,0];
 339
 340 /* 49 Riccati */
 341 (pn_(49),ans:contrib_ode(eqn:'diff(y,x)=cos(x)-(sin(x)-y)*y,y,x));
 342 [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))];
 343 [method,ode_check(eqn,ans[1])];
 344 [riccati,0];
 345
 346 /* 50 Riccati
 347    Fails as can't solve 'diff(z,x,2) - sin(2*x)*'diff(z,x) + cos(2*x)*z = 0
 348 */
 349 (pn_(50),ans:contrib_ode(eqn:'diff(y,x)=cos(2*x)+(sin(2*x)+y)*y,y,x));
 350 [[y=-'diff(%u,x,1)/%u,-'diff(%u,x,1)*sin(2*x)+%u*cos(2*x)+'diff(%u,x,2) = 0]];
 351
 352 /* 51 Riccati */
 353
 354 /* 52 Riccati */
 355 (pn_(52),ans:contrib_ode(eqn:'diff(y,x)=(3+x-4*y)^2,y,x));
 356 [y = (x*(2*%c*%e^(4*x)+2)+5*%c*%e^(4*x)+7)/(8*%c*%e^(4*x)+8) ];
 357 [method,ode_check(eqn,ans[1])];
 358 [riccati,0];
 359
 360 /* 53 Riccati */
 361 (pn_(53),ans:contrib_ode(eqn:'diff(y,x)=(1+4*x+9*y)^2,y,x));
 362  [y = -((x*(12*%c*%e^(12*%i*x)+12)+(2*%i*%c+3*%c)*%e^(12*%i*x)-2*%i+3)
 363           /(27*%c*%e^(12*%i*x)+27))];
 364 [method,ode_check(eqn,ans[1])];
 365 [riccati,0];
 366
 367 /* 54 Riccati */
 368
 369 /* 55 Riccati and separable*/
 370 assume(a>0,b>0);
 371 [a>0,b>0];
 372 (pn_(55.1),ans:contrib_ode(eqn:'diff(y,x)=a+b*y^2,y,x));
 373 [atan(sqrt(b)*y/sqrt(a))/(sqrt(a)*sqrt(b)) = x+%c];
 374 solve(ans[1],y);
 375 [y = sqrt(a)*tan(sqrt(a)*sqrt(b)*x+%c*sqrt(a)*sqrt(b))/sqrt(b)];
 376 [method,ode_check(eqn,%[1])];
 377 [separable,0];
 378 forget(a>0,b>0);
 379 [a>0,b>0];
 380
 381 assume(a>0,b<0);
 382 [a>0,b<0];
 383 (pn_(55.2),ans:contrib_ode(eqn:'diff(y,x)=a+b*y^2,y,x));
 384 [log(-((sqrt(a)*sqrt(-b)-b*y)/(b*y+sqrt(a)*sqrt(-b))))/(2*sqrt(a)*sqrt(-b))=x+%c];
 385 solve(%[1],y);
 386 [y=-((sqrt(a)*sqrt(-b)*%e^(2*sqrt(a)*sqrt(-b)*x+2*%c*sqrt(a)*sqrt(-b))+sqrt(a)*sqrt(-b))/(b*%e^(2*sqrt(a)*sqrt(-b)*x+2*%c*sqrt(a)*sqrt(-b))-b))];
 387 [method,ode_check(eqn,%[1])];
 388 [separable,0];
 389 forget(a>0,b<0);
 390 [a>0,b<0];
 391
 392 /* 57 Riccati */
 393
 394 /* 58 Riccati */
 395
 396 /* 61 Riccati */
 397
 398 /* 62 Riccati - See Kamke 1.27 */
 399 /* (pn_(62),ans:contrib_ode(eqn:'diff(y,x)=1+a*(x-y)*y,y,x)); */
 400
 401 /* 63 Riccati */
 402
 403 /* 64 */
 404 (pn_(64),ans:contrib_ode(eqn:'diff(y,x)=x*y*(3+y),y,x));
 405 [-((log(y+3)-log(y))/3)=x^2/2+%c];
 406 map(exp,3*ans[1]);
 407 y/(y+3) = %e^(3*(x^2/2+%c));
 408 ans:first(solve(%,y));
 409 y=-3*%e^(3*x^2/2+3*%c)/(%e^(3*x^2/2+3*%c)-1);
 410 [method,ode_check(eqn,%)];
 411 [separable,0];
 412
 413 /* 65 Riccati */
 414 (pn_(65),ans:contrib_ode(eqn:'diff(y,x)=1-x-x^3+(1+2*x^2)*y-x*y^2,y,x));
 415 [y = (x^2*%e^x+%e^x+x*(%c-%e^x))/(x*%e^x-%e^x+%c) ];
 416 [method,ode_check(eqn,ans[1])];
 417 [riccati,0];
 418
 419 /* 66 Riccati */
 420 (pn_(66),ans:contrib_ode(eqn:'diff(y,x)=x*(2+x^2*y-y^2),y,x));
 421 [y = %e^-(x^4/4)*(x^2*%e^(x^4/4)*(sqrt(%pi)*%c*erf(x^2/2)+sqrt(%pi))
 422      +2*%c)/(sqrt(%pi)*%c*erf(x^2/2)+sqrt(%pi))];
 423 [method,ode_check(eqn,ans[1])];
 424 [riccati,0];
 425
 426 /* 67 Riccati */
 427 (pn_(67),ans:contrib_ode(eqn:'diff(y,x)=x+(1-2*x)*y-(1-x)*y^2,y,x));
 428 [y = (%e^x+%c*x+%c)/(%e^x+%c*x)];
 429 [method,ode_check(eqn,ans[1])];
 430 [riccati,0];
 431
 432 /* 68 */
 433 (pn_(68),ans:contrib_ode(eqn:'diff(y,x)=a*x*y^2,y,x));
 434 [-1/(a*y)=x^2/2+%c];
 435 ans:first(solve(ans[1],y));
 436 y=-2/(a*x^2+2*%c*a);
 437 [method,ode_check(eqn,ans)];
 438 [separable,0];
 439
 440 /* 70 Riccati */
 441
 442 /* 72 Riccati */
 443 (pn_(72),ans:contrib_ode(eqn:'diff(y,x)=sin(x)*(2*sec(x)^2-y^2),y,x));
 444 [[y='diff(%u,x,1)/(%u*sin(x)),2*%u*sec(x)^2*sin(x)^3-'diff(%u,x,2)*sin(x)+'diff(%u,x,1)*cos(x)=0]];
 445
 446 /* 73 Riccati - FIXME 2006-12-11 - Runs forever */
 447 /* (pn_(73),ans:contrib_ode(eqn:'diff(y,x)+4*csc(x)=(3-cot(x))*y+sin(x)*y^2,y,x));
 448 [[y=-'diff(%u,x,1)/(%u*sin(x)),-4*%u*csc(x)*sin(x)^2-'diff(%u,x,1)*((3-cot(x))*sin(x)+cos(x))+'diff(%u,x,2)*sin(x)=0]]; */
 449
 450 /* 74 */
 451 (pn_(74),ans:contrib_ode(eqn:'diff(y,x)=y*sec(x)+(sin(x)-1)^2,y,x));
 452 [y = (tan(x)+sec(x))*(-8*(log(sin(x)^2/(cos(x)+1)^2+1)/4
 453                                -log(sin(x)/(cos(x)+1)+1)/2
 454                                +sin(x)/((cos(x)+1)
 455                                        *(2*sin(x)^2/(cos(x)+1)^2+2)))
 456                            +8*(-log(sin(x)^2/(cos(x)+1)^2+1)/8
 457                               +log(sin(x)/(cos(x)+1)+1)/4
 458                               -((sin(x)^3/(cos(x)+1)^3-sin(x)^2/(cos(x)+1)^2
 459                                                      +sin(x)/(cos(x)+1))
 460                                /(4*sin(x)^4/(cos(x)+1)^4
 461                                 +8*sin(x)^2/(cos(x)+1)^2+4)))
 462                            +2*(log(sin(x)/(cos(x)+1)+1)
 463                               -log(sin(x)^2/(cos(x)+1)^2+1)/2)+%c)];
 464 /* y = (tan(x)+sec(x))/(%c+sin(x)) ??? */
 465 [method,ode_check(eqn,ans[1])];
 466 [linear,0];
 467
 468 /* 75 */
 469 (pn_(75),ans:contrib_ode(eqn:'diff(y,x)+tan(x)*(1-y^2)=0,y,x));
 470 [-((log(y+1)-log(y-1))/2) = log(sec(x))+%c];
 471 map(exp,2*ans[1]);
 472 (y-1)/(y+1)=%e^(2*(log(sec(x))+%c));
 473 ans:first(solve(%,y));
 474 y=-((%e^(2*%c)*sec(x)^2+1)/(%e^(2*%c)*sec(x)^2-1));
 475 [method,ode_check(eqn,ans)];
 476 [separable,0];
 477
 478 /* 76 Riccati */
 479
 480 /* 78 Abel */
 481 (pn_(78),ans:contrib_ode(eqn:'diff(y,x)+(a*x+y)*y^2=0,y,x));
 482 false;
 483
 484 /* 79 Abel */
 485 (pn_(79),ans:contrib_ode(eqn:'diff(y,x)+(a*exp(x)+y)*y^2=0,y,x));
 486 false;
 487
 488 /* 88 */
 489 (pn_(88),ans:contrib_ode(eqn:'diff(y,x)+2*x*y*(1+a*x^2*y^2)=0,y,x));
 490 [y=%e^-x^2/sqrt(%c-a*(2*x^2+1)*%e^-(2*x^2)/2)];
 491 [method,ode_check(eqn,ans[1])];
 492 [bernoulli,0];
 493
 494 /* 91 */
 495 (pn_(91),ans:contrib_ode(eqn:'diff(y,x)+(tan(x)+y^2*sec(x))*y=0,y,x));
 496 [y=1/(sec(x)*sqrt(2*sin(x)+%c))];
 497 [method,ode_check(eqn,ans[1])];
 498 [bernoulli,0];
 499
 500 /* 92 */
 501 (pn_(92),ans:contrib_ode(eqn:'diff(y,x)+y^3*sec(x)*tan(x)=0,y,x));
 502 [1/(2*y^2)=1/cos(x)+%c];
 503 ans:first(solve(ans[1],y));
 504 y=-sqrt(cos(x)/(%c*cos(x)+1))/sqrt(2);
 505 [method,ode_check(eqn,ans)];
 506 [separable,0];
 507
 508 /* 93 - Generic Abel equation */
 509
 510 /* 94 */
 511 (pn_(94),ans:contrib_ode(eqn:'diff(y,x)=(tan(x)+y^3*sec(x))*y,y,x));
 512 [y=sec(x)/(%c-3*(tan(x)^3/3+tan(x)))^(1/3)];
 513 [method,ode_check(eqn,ans[1])];
 514 [bernoulli,0];
 515
 516 /* 95 */
 517 /* (pn_(95),ans:contrib_ode(eqn:'diff(y,x)=a*x^(n/(1-n))+b*y^n,y,x)); */
 518
 519 /* 96 - Too general */
 520 /* 97 - Too general */
 521 /* 98 - Too general */
 522
 523 /* 99 */
 524 (pn_(99),ans:contrib_ode(eqn:'diff(y,x)=sqrt(y),y,x));
 525 [2*sqrt(y)=x+%c];
 526 [method,ode_check(eqn,ans[1])];
 527 [separable,0];
 528
 529 /* 100 */
 530 (pn_(100),ans:contrib_ode(eqn:'diff(y,x)=a+by+sqrt(c+d*y),y,x));
 531 [((-2*by-2*a)*log(sqrt(d*y+c)+by+a)+2*sqrt(d*y+c))/d = x+%c];
 532 [method,ode_check(eqn,ans[1])];
 533 [separable,0];