share/contrib/diffequations/tests/rtestode_murphy_2_1.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Murphy equations 2.1 - 2.100
   5
   6   Reference:
   7     G M Murphy, Ordinary Differential Equations and Their
   8     Solutions, Van Nostrand, 1960, pp 15-23
   9
  10     First Order and Second or Higher Degree, p278 ff
  11 */
  12
  13 /* Don't kill(all); It messes up trigsimp */
  14
  15 /* Eliminate parameter t from singular solution */
  16 (elim_(ans,x,y,t):=block([s],s:first(solve(ans[2],t)),s:solve(ev(ans[1],s),y),s[1]),done);
  17 done;
  18
  19 /* Print ode number*/
  20 (pn_(n_):=print("Murphy ODE 2.",n_),true);
  21 true;
  22
  23 /* 1 */
  24 assume(not(equal(n,-2)));
  25 '[not(equal(n, -2))];
  26 (pn_(1),ans:contrib_ode(eqn:'diff(y,x)^2=a*x^n,y,x));
  27 [-(((n+2)*y+2*x*sqrt(a*x^n))/(n+2))=%c];
  28 solve(ans[1],y);
  29 [y=-((2*x*sqrt(a*x^n)+%c*n+2*%c)/(n+2))];
  30 [method,ode_check(eqn,%[1])];
  31 [lie,0];
  32 forget(not(equal(n,-2)));
  33 '[not(equal(n,-2))];
  34
  35 /* 2 */
  36 (pn_(2),ans:contrib_ode(eqn:'diff(y,x)^2=y,y,x));
  37 [4*y-x^2+2*%c*x-%c^2=0];
  38 solve(ans[1],y);
  39 [y=(x^2-2*%c*x+%c^2)/4];
  40 [method,ode_check(eqn,%[1])];
  41 [lagrange,0];
  42
  43 /* 3 */
  44 (pn_(3),ans:contrib_ode(eqn:'diff(y,x)^2=x-y,y,x));
  45 [[x=%c-2*(log(%t-1)+%t),y=x-%t^2]]; /* Correct */
  46 method;
  47 lagrange;
  48
  49 /* 4 Very long "answer" from ode1_lie */
  50 /* (pn_(4),ans:contrib_ode(eqn:'diff(y,x)^2=x^2+y,y,x)); */
  51
  52 /* 5 Very long "answer" from ode1_lie */
  53 /* (pn_(5),ans:contrib_ode(eqn:'diff(y,x)^2+x^2=4*y,y,x)); */
  54
  55 /* 6 Very long "answer" from ode1_lie */
  56 /* (pn_(6),ans:contrib_ode(eqn:'diff(y,x)^2+3*x^2=8*y,y,x)); */
  57
  58 /* 7 ode1_lie takes forever */
  59 /* (pn_(7),ans:contrib_ode(eqn:'diff(y,x)^2+a*x^2+b*y=0,y,x)); */
  60
  61 /* 8 */ /* CHECKME */
  62 (pn_(8),ans:contrib_ode(eqn:'diff(y,x)^2=1+y^2,y,x));
  63 [[x=%c-log(2*sqrt(%t^2-1)+2*%t),y=-sqrt(%t^2-1)],[x=log(2*sqrt(%t^2-1)+2*%t)+%c,y=sqrt(%t^2-1)]];
  64 method;
  65 lagrange;
  66
  67 /* 9 */
  68 (pn_(9),ans:contrib_ode(eqn:'diff(y,x)^2=1-y^2,y,x));
  69 [[x = asin(%t)+%c,y = -sqrt(1-%t^2)],[x=%c-asin(%t),y=sqrt(1-%t^2)]];
  70 solve(ans[1][1],%t);
  71 [%t = sin(x-%c)];
  72 ans:ans[1][2],%;
  73 y = -sqrt(1-sin(x-%c)^2); /* y = sin(x+%c) */
  74 [method,ode_check(eqn,ans)];
  75 [lagrange,0];
  76 /* ans[2] has %c -> -%c */
  77 /* Other solutions y=1, y=-1 */
  78
  79 /* 10 */
  80 (pn_(10),ans:contrib_ode(eqn:'diff(y,x)^2=a^2-y^2,y,x));
  81 [[x=asin(%t/abs(a))+%c,y=-sqrt(a^2-%t^2)],[x=%c-asin(%t/abs(a)),y=sqrt(a^2-%t^2)]];
  82 solve(ans[1][1],%t);
  83 [%t = abs(a)*sin(x-%c)];
  84 ans:ans[1][2],%;
  85 y=-sqrt(a^2-a^2*sin(x-%c)^2);
  86 [method,ode_check(eqn,ans)];
  87 [lagrange,0];
  88 /* Solutions in Murphy are
  89      y=(a*(1-%c^2)*sin(x)+2*%c*a*cos(x))/(1+%c^2))
  90      y+a*sin(x)=0
  91      y=a, y=-a
  92 */
  93
  94 /* 11 */
  95 (pn_(11),ans:contrib_ode(eqn:'diff(y,x)^2=a^2*y^2,y,x));
  96 [y = %c*%e^(a*x),y = %c*%e^-(a*x)];
  97 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
  98 [factor,0,0];
  99
 100 /* 12 */
 101 assume(a>0,b>0);
 102 '[a>0,b>0];
 103 (pn_(12),ans:contrib_ode(eqn:'diff(y,x)^2=a+b*y^2,y,x));
 104 [[x=%c-log(2*sqrt(%t^2-a)+2*%t)/sqrt(b),y=-sqrt(%t^2-a)/sqrt(b)],[x=log(2*sqrt(%t^2-a)+2*%t)/sqrt(b)+%c,y=sqrt(%t^2-a)/sqrt(b)]]; /* CHECKME */
 105 method;
 106 lagrange;
 107 forget(a>0,b>0);
 108 '[a>0,b>0];
 109
 110 /* 13 */
 111 (pn_(13),ans:contrib_ode(eqn:'diff(y,x)^2=x^2*y^2,y,x));
 112 [y = %c*%e^(x^2/2),y = %c*%e^-(x^2/2)];
 113 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 114 [factor,0,0];
 115
 116 /* 14 FIXME: crashes ode1_lagrange */
 117 /* (pn_(14),ans:contrib_ode(eqn:'diff(y,x)^2=(y-1)*y^2,y,x)); */
 118
 119 /* 15 ode1_lagrange takes "forever" */
 120 /* (pn_(15),ans:contrib_ode(eqn:'diff(y,x)^2=(y-a)*(y-b)*(y-c),y,x)); */
 121
 122 /* 16 ode1_lagrange returns a "solution" */
 123 /* (pn_(16),ans:contrib_ode(eqn:'diff(y,x)^2=a^2*y^n,y,x)); */
 124
 125 /* 17 ode1_lie crashes */
 126 /* (pn_(17),ans:contrib_ode(eqn:'diff(y,x)^2=a^2*(1-log(y)^2)*y^2,y,x)); */
 127
 128 /* 18 to 24 are very (too?) general */
 129
 130 /* 25 */
 131 (pn_(25),ans:contrib_ode(eqn:'diff(y,x)^2+2*'diff(y,x)+x=0,y,x));
 132 [-((3*y+sqrt(1-x)*(2*x-2)+3*x)/3) = %c];
 133 solve(ans[1],y);
 134 [y = -(((2*sqrt(1-x)+3)*x-2*sqrt(1-x)+3*%c)/3)];
 135 [method,ode_check(eqn,%)];
 136 [lie,0];
 137
 138 /* 26 */
 139 (pn_(26),ans:contrib_ode(eqn:'diff(y,x)^2-2*'diff(y,x)+a*(x-y)=0,y,x),0);
 140 0$
 141 [method,ode_check(eqn,ans[1])];
 142 [lie,0];
 143 /* Murphy gives a*(C+x)^2=4*(C+y), a*y=1+a*x */
 144
 145 /* 27 */
 146 (pn_(27),ans:contrib_ode(eqn:'diff(y,x)^2-2*'diff(y,x)-y^2=0,y,x));
 147 [[x=-log(2*sqrt(%t^2-2*%t)+2*%t-2)+sqrt(%t^2-2*%t)/%t+%c,y=-sqrt(%t^2-2*%t)],[x=log(2*sqrt(%t^2-2*%t)+2*%t-2)-sqrt(%t^2-2*%t)/%t+%c,y=sqrt(%t^2-2*%t)]];
 148 method; /* CHECKME */
 149 lagrange;
 150
 151 /* 28 */
 152 (pn_(28),ans:contrib_ode(eqn:'diff(y,x)^2-5*'diff(y,x)+6=0,y,x));
 153 [y=3*x+%c,y=2*x+%c];
 154 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 155 [factor,0,0];
 156
 157 /* 29 */
 158 (pn_(29),ans:contrib_ode(eqn:'diff(y,x)^2-7*'diff(y,x)+12=0,y,x));
 159 [y=4*x+%c,y=3*x+%c];
 160 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 161 [factor,0,0];
 162
 163
 164 /* 30 */
 165 (pn_(30),ans:contrib_ode(eqn:'diff(y,x)^2+a*'diff(y,x)+b=0,y,x));
 166 [-((2*y+sqrt(a^2-4*b)*x+a*x)/2) = %c];
 167 solve(ans[1],y);
 168 [y = -(((sqrt(a^2-4*b)+a)*x+2*%c)/2)];
 169 [method,ode_check(eqn,%)];
 170 [lie,0];
 171
 172 /* 31 */
 173 (pn_(31),ans:contrib_ode(eqn:'diff(y,x)^2+a*'diff(y,x)+b*x=0,y,x));
 174 [-((12*b*y+sqrt(a^2-4*b*x)*(4*b*x-a^2)+6*a*b*x)/(12*b)) = %c];
 175 solve(ans[1],y);
 176 [y = -((sqrt(a^2-4*b*x)*(4*b*x-a^2)+6*a*b*x+12*%c*b)/(12*b))];
 177 [method,ode_check(eqn,%)];
 178 [lie,0];
 179
 180 /* 32 */
 181 (pn_(32),ans:contrib_ode(eqn:'diff(y,x)^2+a*'diff(y,x)+b*y=0,y,x));
 182 [[x=%c-((log(%t)*a+2*%t)/b),y = -((%t*a+%t^2)/b)]]; /* Correct */
 183 method;
 184 lagrange;
 185
 186 /* 33 */
 187 (pn_(33),ans:contrib_ode(eqn:'diff(y,x)^2+x*'diff(y,x)+1=0,y,x));
 188 [-((4*y-4*log(2*sqrt(x^2-4)+2*x)+x*sqrt(x^2-4)+x^2)/4)=%c];
 189 solve(ans[1],y);
 190 [y=(4*log(2*sqrt(x^2-4)+2*x)-x*sqrt(x^2-4)-x^2-4*%c)/4];
 191 [method,ode_check(eqn,%)];
 192 [lie,0];
 193
 194 /* 34 */
 195 (pn_(34),ans:contrib_ode(eqn:'diff(y,x)^2+x*'diff(y,x)-y=0,y,x));
 196 [y=%c*x+%c^2,y=-x^2/4];
 197 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 198 [clairaut,0,0];
 199
 200 /* 35 */
 201 (pn_(35),ans:contrib_ode(eqn:'diff(y,x)^2-x*'diff(y,x)+y=0,y,x));
 202 [y=%c*x-%c^2,y=x^2/4];
 203 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 204 [clairaut,0,0];
 205
 206 /* 36 */
 207 (pn_(36),ans:contrib_ode(eqn:'diff(y,x)^2-x*'diff(y,x)-y=0,y,x));
 208 [[x=%e^-(log(%t)/2)*(2*%e^(3*log(%t)/2)/3+%c),y=%t^2-%t*x]];
 209 ans:radcan(%);
 210 [[x=(2*%t^(3/2)+3*%c)/(3*sqrt(%t)),y=%t^2-%t*x]];
 211 ans:ans,3*%c=%c; /* change constant */
 212 [[x=(2*%t^(3/2)+%c)/(3*sqrt(%t)),y=%t^2-%t*x]];
 213 ans[1][2]:ans[1][2],ans[1][1];
 214 y=%t^2-sqrt(%t)*(2*%t^(3/2)+%c)/3;
 215 expand(3*ans);
 216 [[3*x = 2*%t+%c/sqrt(%t),3*y = %t^2-%c*sqrt(%t)]];  /* correct */
 217 method;
 218 lagrange;
 219
 220 /* 37 */
 221 (pn_(37),ans:contrib_ode(eqn:'diff(y,x)^2+x*'diff(y,x)+x-y=0,y,x));
 222 [[x=%e^-%t*(%c-2*(%t-1)*%e^%t),y = (%t+1)*x+%t^2]];
 223 ans[1][2]:ans[1][2],ans[1][1];
 224 y=(%t+1)*%e^-%t*(%c-2*(%t-1)*%e^%t)+%t^2;
 225 expand(ans);
 226 [[x=%c*%e^-%t-2*%t+2,y=%c*%t*%e^-%t+%c*%e^-%t-%t^2+2]]; /* correct */
 227 method;
 228 lagrange;
 229
 230 /* 38 */
 231 (pn_(38),ans:contrib_ode(eqn:'diff(y,x)^2+(1-x)*'diff(y,x)+y=0,y,x));
 232 [y=%c*x-%c-%c^2,y=(x^2-2*x+1)/4];
 233 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 234 [clairaut,0,0];
 235
 236 /* 39 */
 237 (pn_(39),ans:contrib_ode(eqn:'diff(y,x)^2-(1+x)*'diff(y,x)+y=0,y,x));
 238 [y=%c*x-%c^2+%c,y=(x^2+2*x+1)/4];
 239 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 240 [clairaut,0,0];
 241
 242 /* 40 */
 243 (pn_(40),ans:contrib_ode(eqn:'diff(y,x)^2-(2-x)*'diff(y,x)+1-y=0,y,x));
 244 [y=%c*x+%c^2-2*%c+1,y=-((x^2-4*x)/4)];
 245 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 246 [clairaut,0,0];
 247
 248 /* 41 */
 249 (pn_(41),ans:contrib_ode(eqn:'diff(y,x)^2+(a+x)*'diff(y,x)-y=0,y,x));
 250 [y=%c*x+%c*a+%c^2,y=-((x^2+2*a*x+a^2)/4)];
 251 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 252 [clairaut,0,0];
 253
 254 /* 42 */
 255 (pn_(42),ans:contrib_ode(eqn:'diff(y,x)^2-2*x*'diff(y,x)+1=0,y,x));
 256 [-((2*y-log(2*sqrt(x^2-1)+2*x)+x*sqrt(x^2-1)-x^2)/2)=%c];
 257 solve(ans[1],y);
 258 [y=(log(2*sqrt(x^2-1)+2*x)-x*sqrt(x^2-1)+x^2-2*%c)/2];
 259 [method,ode_check(eqn,%)];
 260 [lie,0];
 261
 262 /* 43 */
 263 (pn_(43),ans:contrib_ode(eqn:'diff(y,x)^2+2*x*'diff(y,x)-3*x^2=0,y,x));
 264 [y = x^2/2+%c,y = %c-3*x^2/2];
 265 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 266 [factor,0,0];
 267
 268 /* 44 */
 269 (pn_(44),ans:contrib_ode(eqn:'diff(y,x)^2*x*'diff(y,x)-y=0,y,x));
 270 [[x = %c*%e^-(3*log(%t^2-1)/2),y = %t^3*x]]; /* parametric soln */
 271 first(solve(ans[1][1]^2,%t));
 272 %t=-sqrt(x^(2/3)+%c^(2/3))/x^(1/3);
 273 ans:ev(ans[1][2],%);
 274 y=-(x^(2/3)+%c^(2/3))^(3/2);
 275 [method,ode_check(eqn,ans)];
 276 [lagrange,0];
 277
 278 /* 45 */
 279 (pn_(45),ans:contrib_ode(eqn:'diff(y,x)^2+2*x*diff(y,x)-y=0,y,x));
 280 [4*y-x^2+2*%c*x-%c^2=0];
 281 ans:solve(ans[1],y);
 282 [y = (x^2-2*%c*x+%c^2)/4];
 283 [method,ode_check(eqn,ans[1])];
 284 [lagrange,0];
 285
 286 /* 46 */
 287 (pn_(46),ans:contrib_ode(eqn:'diff(y,x)^2-2*x*'diff(y,x)+2*y=0,y,x));
 288 [y=(2*%c*x-%c^2)/2,y=x^2/2];
 289 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 290 [clairaut,0,0];
 291
 292 /* 47 */
 293 (pn_(47),ans:contrib_ode(eqn:'diff(y,x)^2+(1+2*x)*diff(y,x)-x*(1-x)=0,y,x));
 294 [-((8*y+asin(2*x-1)+(4*x-2)*sqrt(x-x^2))/8) = %c];
 295 solve(ans[1],y);
 296 [y = -((asin(2*x-1)+(4*x-2)*sqrt(x-x^2)+8*%c)/8)];
 297 [method,radcan(ode_check(eqn,%))];
 298 [lie,0];
 299
 300 /* 48 */
 301 (pn_(48),ans:contrib_ode(eqn:'diff(y,x)^2+(1-x)*diff(y,x)-2*(x-y)=0,y,x));
 302 [-((2*sqrt(x-y)-sqrt(2)*log(sqrt(2)*sqrt(x-y)+1)-sqrt(2)*x)/sqrt(2))=%c];
 303 [method,ode_check(eqn,ans[1])];
 304 [lie,0];
 305
 306 /* 49 */
 307 (pn_(49),ans:contrib_ode(eqn:'diff(y,x)^2+3*x*'diff(y,x)-y=0,y,x));
 308 [[x = %e^-(3*log(%t)/2)*(%c-2*%e^(5*log(%t)/2)/5),y = 3*%t*x+%t^2]]; /* CHECKME */
 309 method;
 310 lagrange;
 311
 312 /* 50 */
 313 (pn_(50),ans:contrib_ode(eqn:'diff(y,x)^2-4*(1+x)*'diff(y,x)+4*y=0,y,x));
 314 [y=(4*%c*x-%c^2+4*%c)/4,y=x^2+2*x+1];
 315 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 316 [clairaut,0,0];
 317
 318 /* 51 */
 319 assume(4*c+a^2>0,b>0);
 320 [4*c+a^2 > 0,b > 0];
 321 (pn_(51),ans:contrib_ode(eqn:'diff(y,x)^2+a*x*'diff(y,x)=b+c*x^2,y,x));
 322 [-((sqrt(4*c+a^2)*(4*y+a*x^2)+4*b*asinh(sqrt(4*c+a^2)*x/(2*sqrt(b)))+sqrt(4*c+a^2)*x*sqrt((4*c+a^2)*x^2+4*b))/(4*sqrt(4*c+a^2)))=%c];
 323 solve(ans[1],y);
 324 [y = -((4*b*asinh(sqrt(4*c+a^2)*x/(2*sqrt(b)))+sqrt(4*c+a^2)*x*sqrt(4*c*x^2+a^2*x^2+4*b)+a*sqrt(4*c+a^2)*x^2+4*%c*sqrt(4*c+a^2))/(4*sqrt(4*c+a^2)))];
 325 [method,ode_check(eqn,%[1])];
 326 [lie,0];
 327 forget(4*c+a^2>0,b>0);
 328 [4*c+a^2>0,b>0];
 329
 330 /* 52 */
 331 (pn_(52),ans:contrib_ode(eqn:'diff(y,x)^2-a*x*'diff(y,x)+a*y=0,y,x));
 332 [y=(%c*a*x-%c^2)/a,y=a*x^2/4];
 333 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 334 [clairaut,0,0];
 335
 336 /* 53 ode1_lie takes forever */
 337 /* (pn_(53),ans:contrib_ode(eqn:'diff(y,x)^2+a*x*'diff(y,x)+b*x^2+c*y=0,y,x));*/
 338
 339 /* 54 */
 340 (pn_(54),ans:contrib_ode(eqn:'diff(y,x)^2+(a+b*x)*'diff(y,x)+c=b*y,y,x));
 341 [y=(%c*b*x+c+%c*a+%c^2)/b,y=-((b^2*x^2+2*a*b*x-4*c+a^2)/(4*b))];
 342 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 343 [clairaut,0,0];
 344
 345 /* 55 ode1_lie takes forever */
 346 /*(pn_(55),ans:contrib_ode(eqn:'diff(y,x)^2-2*x^2*'diff(y,x)+2*x*y=0,y,x));*/
 347
 348 /* 56 ode1_lie takes forever */
 349 /*(pn_(56),ans:contrib_ode(eqn:'diff(y,x)^2-a*x^2*'diff(y,x)+b*x*y=0,y,x));*/
 350
 351 /* 57 */
 352 (pn_(57),ans:contrib_ode(eqn:'diff(y,x)^2-a*x^3*'diff(y,x)+2*a*x^2*y=0,y,x));
 353 [-((sqrt(a)*sqrt(a^2*x^4-8*a*y)-a^(3/2)*x^2)/(sqrt(3)*a)) = %c];
 354 solve(ans[1],sqrt(a^2*x^4-8*a*y));
 355 [sqrt(a^2*x^4-8*a*y) = (a^(3/2)*x^2-sqrt(3)*%c*a)/sqrt(a)];
 356 %[1]^2;
 357 a^2*x^4-8*a*y = (a^(3/2)*x^2-sqrt(3)*%c*a)^2/a;
 358 ans:solve(%,y);
 359 [y=(2*sqrt(3)*%c*a^(5/2)*x^2-3*%c^2*a^2)/(8*a^2)];
 360 [method,ode_check(eqn,ans[1])];
 361 [lie,0];
 362
 363 /* 58 ode1_lie takes forever */
 364 /* (pn_(58),ans:contrib_ode(eqn:'diff(y,x)^2-a*x^3*'diff(y,x)+4*a*x^2*y=0,y,x));*/
 365
 366 /* 59 */
 367 (pn_(59),ans:contrib_ode(eqn:'diff(y,x)^2+4*x^5*'diff(y,x)-12*x^4*y=0,y,x),0);
 368 0$
 369 [method,ode_check(eqn,ans[1])];
 370 [lie,0];
 371
 372 /* 60 */
 373 (pn_(60),ans:contrib_ode(eqn:'diff(y,x)^2+2*'diff(y,x)*cosh(x)+1=0,y,x));
 374 [-%e^-x*(2*%e^x*y+2*%e^x*sinh(x)+%e^(2*x)+1)/2 = %c];
 375 solve(ans[1],y);
 376 [y = -%e^-x*(2*%e^x*sinh(x)+%e^(2*x)+2*%c*%e^x+1)/2];
 377 [method,expand(exponentialize(ode_check(eqn,%[1])))];
 378 [lie,0];
 379
 380 /* 61 */
 381 (pn_(61),ans:contrib_ode(eqn:'diff(y,x)^2+y*'diff(y,x)=x*(x+y),y,x));
 382 [y = x^2/2+%c,y = %e^-x*(%c-(x-1)*%e^x)];
 383 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 384 [factor,0,0];
 385
 386 /* 62 */
 387 (pn_(62),ans:contrib_ode(eqn:'diff(y,x)^2-y*'diff(y,x)+exp(x)=0,y,x));
 388 [2*log(sqrt(y^2-4*%e^x)+y)=%c];
 389 map(exp,ans[1]/2);
 390 sqrt(y^2-4*%e^x)+y=%e^(%c/2);
 391 %,exp(%c/2)=%c;
 392 sqrt(y^2-4*%e^x)+y=%c;
 393 (%-y)^2;
 394 y^2-4*%e^x=(%c-y)^2;
 395 solve(expand(%),y);
 396 [y=(4*%e^x+%c^2)/(2*%c)];
 397 [method,ode_check(eqn,%[1])];
 398 [lie,0];
 399 /* FIXME? y=2*sqrt(exp(x)) and y=-2*sqrt(exp(x)) are singular solutions */
 400
 401 /* 63 */
 402 (pn_(63),ans:contrib_ode(eqn:'diff(y,x)^2+(x+y)*'diff(y,x)+x*y=0,y,x));
 403 [y = %c-x^2/2,y = %c*%e^-x];
 404 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 405 [factor,0,0];
 406
 407 /* [[x = %c-log(%t),y = -%t]];
 408 ans[1][1],solve(ans[1][2],%t);
 409 x=%c-log(-y);
 410 solve(%,y);
 411 [y=-%e^(%c-x)];
 412 [method,ode_check(eqn,%[1])];
 413 [lagrange,0]; */
 414 /* FIXME: y=(%c-x^2)/2 also a solution */
 415
 416 /* 64 */
 417 (pn_(64),ans:contrib_ode(eqn:'diff(y,x)^2+2*y*'diff(y,x)-2*x=0,y,x));
 418 false;
 419
 420 /* 65 */
 421 (pn_(65),ans:contrib_ode(eqn:'diff(y,x)^2+(1+2*y)*'diff(y,x)+y*(y-1)=0,y,x));
 422 [[x = 2*(log(sqrt(1-8*%t)-1)/2-(3*log(sqrt(1-8*%t)+1))/2)+%c,
 423   y = -((2*%t+sqrt(1-8*%t)-1)/2)],
 424  [x = 2*(log(sqrt(1-8*%t)+1)/2-(3*log(sqrt(1-8*%t)-1))/2)+%c,
 425   y = ((-2*%t)+sqrt(1-8*%t)+1)/2]]$
 426 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 427 [lagrange,0,0];
 428
 429 /* 66 */
 430 (pn_(66),ans:contrib_ode(eqn:'diff(y,x)^2-2*(x-y)*'diff(y,x)-4*x*y=0,y,x));
 431 [y=x^2+%c,y=%c*%e^-(2*x)];
 432 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 433 [factor,0,0];
 434
 435 /* 67 */
 436 (pn_(67),ans:contrib_ode(eqn:'diff(y,x)^2-(1+4*y)*'diff(y,x)-(1+4*y)*y=0,y,x));
 437 [[x=%c-((-asinh(%t/abs(%t)-1/(4*abs(%t)))+8*asinh((64*%t-8)/8)/2^(5/2)+log(%t))/2),
 438   y=-((sqrt(32*%t^2-8*%t+1)+4*%t+1)/8)],
 439  [x=%c-((asinh(%t/abs(%t)-1/(4*abs(%t)))-8*(asinh((64*%t-8)/8)/2^(5/2))+log(%t))/2),
 440   y = ((sqrt(32*%t^2-8*%t+1)-4*%t-1)/8)]];
 441 [method,radcan(ode_check(eqn,ans[1])),radcan(ode_check(eqn,ans[2]))];
 442 [lagrange,0,0];
 443
 444 /* 68 */
 445 (pn_(68),ans:contrib_ode(eqn:'diff(y,x)^2-2*(1-3*y)*'diff(y,x)-(4-9*y)*y=0,y,x));
 446 [[x = (sqrt(2)*(-(log((2*sqrt(3*%t+2)-2^(3/2))
 447                               /(2*sqrt(3*%t+2)+2^(3/2)))
 448                       /2^(3/2))-log(3*%t)/sqrt(2)))/3+%c,
 449   y = -((sqrt(2)*sqrt(3*%t+2)+3*%t-2)/9)],
 450  [x = (sqrt(2)*(log((2*sqrt(3*%t+2)-2^(3/2))/(2*sqrt(3*%t+2)+2^(3/2)))
 451                       /2^(3/2)-log(3*%t)/sqrt(2)))/3+%c,
 452   y = (sqrt(2)*sqrt(3*%t+2)-3*%t+2)/9]]$
 453 [method,radcan(ode_check(eqn,ans[1])),radcan(ode_check(eqn,ans[2]))];
 454 [lagrange,0,0];
 455
 456 /* 69 ode1_lie takes forever */
 457 /*(pn_(69),ans:contrib_ode(eqn:'diff(y,x)^2-(a+6*y)*'diff(y,x)+y(3*a+b+9*y)=0,y,x));*/
 458
 459 /* 70 */
 460 (pn_(70),ans:contrib_ode(eqn:'diff(y,x)^2+a*y*'diff(y,x)-a*x=0,y,x));
 461 false;
 462
 463 /* 71 */
 464 (pn_(71),ans:contrib_ode(eqn:'diff(y,x)^2-a*y*'diff(y,x)-a*x=0,y,x));
 465 false;
 466
 467 /* 72 */
 468 (pn_(72),ans:contrib_ode(eqn:'diff(y,x)^2+(a*x+b*y)*'diff(y,x)-a*b*x*y=0,y,x));
 469 false;
 470
 471 /* 73 */
 472 (pn_(73),ans:contrib_ode(eqn:'diff(y,x)^2-x*y*'diff(y,x)+y^2*log(a*y)=0,y,x));
 473 [x-sqrt(x^2-4*log(a*y))=%c];
 474 (ans[1]-x)^2;
 475 x^2-4*log(a*y)=(%c-x)^2;
 476 map(exp,(x^2-%)/4);
 477 a*y=%e^((x^2-(%c-x)^2)/4);
 478 %/a;
 479 y=%e^((x^2-(%c-x)^2)/4)/a;
 480 [method,ode_check(eqn,%)];
 481 [lie,0];
 482 /* FIXME?  there is a second solution */
 483
 484 /* 74 ode1_lie takes forever */
 485 /*(pn_(74),ans:contrib_ode(eqn:'diff(y,x)^2-(1+2*x*y)*'diff(y,x)+y^2*x*y=0,y,x));*/
 486
 487 /* 75 */
 488 (pn_(75),ans:contrib_ode(eqn:'diff(y,x)^2-(4+y^2)*'diff(y,x)+4+y^2=0,y,x));
 489 [[x=1/sqrt(%t-1)+%c,y=-((%t-2)/sqrt(%t-1))],[x=%c-1/sqrt(%t-1),y=(%t-2)/sqrt(%t-1)]];
 490 solve(ans[1][1],sqrt(%t-1));
 491 [sqrt(%t-1) = 1/(x-%c)];
 492 %[1]^2+1;
 493 %t = 1/(x-%c)^2+1;
 494 ans[1][2],%;
 495 y = -((1/(x-%c)^2-1)*abs(x-%c));
 496 %,abs(x-%c)=(x-%c);
 497 y = -((x-%c)*(1/(x-%c)^2-1));
 498 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 499 [lagrange,0,0];
 500
 501 /* 76 ode1_lie takes forever */
 502 /*(pn_(76),ans:contrib_ode(eqn:'diff(y,x)^2-(x-y)*y*'diff(y,x)+x*y^3=0,y,x));*/
 503
 504 /* 77 ode1_lie takes forever */
 505 /*(pn_(77),ans:contrib_ode(eqn:'diff(y,x)^2+x*y^2*'diff(y,x)+y^3=0,y,x));*/
 506
 507 /* 78 very long "answer" from ode1_lie */
 508 /*(pn_(78),ans:contrib_ode(eqn:'diff(y,x)^2-2*x^3*y^2*'diff(y,x)+4*x^2*y^3=0,y,x));*/
 509
 510 /* 79 */
 511 (pn_(79),ans:contrib_ode(eqn:'diff(y,x)^2+x*y*(x^2+y^2)*'diff(y,x)+x^4*y^4=0,y,x));
 512 [y=%c*%e^-(x^4/4),1/(2*y^2)=x^2/2+%c];
 513 solve(ans[2],y);
 514 [y=-1/sqrt(x^2+2*%c),y=1/sqrt(x^2+2*%c)];
 515 [method,ode_check(eqn,ans[1]),ode_check(eqn,%[1]),ode_check(eqn,%[2])];
 516 [factor,0,0,0];
 517
 518 /* 80  very long "answer" from ode1_lie */
 519 /*(pn_(80),ans:contrib_ode(eqn:'diff(y,x)^2+x*y^3*'diff(y,x)+y^4=0,y,x));*/
 520
 521 /* 81 ode1_lie takes forever */
 522 /* (pn_(81),ans:contrib_ode(eqn:'diff(y,x)^2+2*y*'diff(y,x)*cot(x)-y^2=0,y,x));*/
 523
 524 /* 82 ode1_lie takes forever */
 525 /*(pn_(82),ans:contrib_ode(eqn:'diff(y,x)^2-3*x*y^(2/3)*'diff(y,x)+9*y^(5/3)=0,y,x));*/
 526
 527 /* 83 There is a typo in this eqn in Murphy.  This matches the solution given
 528       Very long "answer" from ode1_lie */
 529 /*(pn_(83),ans:contrib_ode(eqn:'diff(y,x)^2+exp(4*x-2*y)*('diff(y,x)-1)=0,y,x));*/
 530
 531 /* 84 */
 532 (pn_(84),ans:contrib_ode(eqn:2*'diff(y,x)^2+x*'diff(y,x)-2*x=0,y,x));
 533 [-((8*y-64*log(2*sqrt(x^2+16*x)+2*x+16)+(x+8)*sqrt(x^2+16*x)+x^2)/8) = %c];
 534 solve(ans[1],y);
 535 [y = (64*log(2*sqrt(x^2+16*x)+2*x+16)+(-x-8)*sqrt(x^2+16*x)-x^2-8*%c)/8];
 536 [method,ode_check(eqn,%)];
 537 [lie,0];
 538
 539 /* 85  */
 540 (pn_(85),ans:contrib_ode(eqn:2*'diff(y,x)^2-(1-x)*'diff(y,x)-y=0,y,x));
 541 [y=%c*x+2*%c^2-%c,y=-((x^2-2*x+1)/8)];
 542 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 543 [clairaut,0,0];
 544
 545 /* 86 Very long "answer" from ode1_lie that doesn't satisfy ode*/
 546 /*(pn_(86),ans:contrib_ode(eqn:2*'diff(y,x)^2-2*x^2*'diff(y,x)+3*x*y=0,y,x));*/
 547
 548 /* 87 */
 549 (pn_(87),ans:contrib_ode(eqn:2*'diff(y,x)^2-2*(6*y-1)*'diff(y,x)+3*y*(6*y-1)=0,y,x));
 550 [[x=2*log(sqrt(1-8*%t)-1)/3+%c,y=-((-4*%t+sqrt(1-8*%t)-1)/12)],[x=2*log(sqrt(1-8*%t)+1)/3+%c,y=(4*%t+sqrt(1-8*%t)+1)/12]];
 551 map(exp,(ans[1][1]-%c)*3/2)+1;
 552 %e^(3*(x-%c)/2)+1 = sqrt(1-8*%t);
 553 %^2;
 554 (%e^(3*(x-%c)/2)+1)^2=1-8*%t;
 555 %,%c=-(2/3)*log(%c); /* a better constant */
 556 (%e^(3*(x+2*log(%c)/3)/2)+1)^2=1-8*%t;
 557 radcan(%);
 558 %c^2*%e^(3*x)+2*%c*%e^(3*x/2)+1 = 1-8*%t;
 559 solve(%,%t);
 560 [%t=-((%c^2*%e^(3*x)+2*%c*%e^(3*x/2))/8)];
 561 ans[1][2],%;
 562 y=-((sqrt(%c^2*%e^(3*x)+2*%c*%e^(3*x/2)+1)+(%c^2*%e^(3*x)+2*%c*%e^(3*x/2))/2-1)/12);
 563 radcan(%);
 564 y=-((%c^2*%e^(3*x)+4*%c*%e^(3*x/2))/24);
 565 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 566 [lagrange,0,0];
 567
 568 /* 88 */
 569 (pn_(88),ans:contrib_ode(eqn:3*'diff(y,x)^2-2*x*'diff(y,x)+y=0,y,x));
 570 [4*y^3-x^2*y^2-18*%c*x*y+4*%c*x^3+27*%c^2 = 0];
 571 method;
 572 lagrange;
 573
 574 /* 89 */
 575 (pn_(89),ans:contrib_ode(eqn:3*'diff(y,x)^2+4*x*'diff(y,x)+x^2-y=0,y,x),0);
 576 0$
 577 [method,ode_check(eqn,ans[1])];
 578 [lie,0];
 579
 580 /* 90 */
 581 (pn_(90),ans:contrib_ode(eqn:4*'diff(y,x)^2=9*x,y,x));
 582 [-y-x^(3/2)=%c];
 583 solve(ans[1],y);
 584 [y=-x^(3/2)-%c];
 585 [method,ode_check(eqn,%)];
 586 [lie,0];
 587
 588 /* 91 Very long "answer" from ode1_lie that doesn't satisfy ode */
 589 /* (pn_(91),ans:contrib_ode(eqn:4*'diff(y,x)^2+2*x*exp(-2*y)*'diff(y,x)-exp(-2*y)=0,y,x));*/
 590
 591 /* 92 */
 592 (pn_(92),ans:contrib_ode(eqn:4*'diff(y,x)^2+2*exp(2*x-2*y)*'diff(y,x)-exp(2*x-2*y)=0,y,x));
 593 false;
 594
 595 /* 93 */
 596 (pn_(93),ans:contrib_ode(eqn:5*'diff(y,x)^2+3*x*'diff(y,x)-y=0,y,x));
 597 [[x=%e^-(3*log(%t)/2)*(%c-2*%e^(5*log(%t)/2)),y=3*%t*x+5*%t^2]];
 598 [method,ode_check(eqn,ans[1])];
 599 [lagrange,0];
 600
 601 /* 94  Very long "answer" from ode1_lie that doesn't satisfy ode */
 602 /* (pn_(94),ans:contrib_ode(eqn:5*'diff(y,x)^2+6*x*'diff(y,x)-2*y=0,y,x)); */
 603
 604 /* 95 ode1_lie takes forever */
 605 /* (pn_(95),ans:contrib_ode(eqn:9*'diff(y,x)^2+3*x*y^4*'diff(y,x)+y^5=0,y,x));*/
 606
 607 /* 96 */
 608 (pn_(96),ans:contrib_ode(eqn:x*'diff(y,x)^2=a,y,x));
 609 [-y-2*sqrt(a/x)*x = %c];
 610 solve(ans[1],y);
 611 [y = -2*sqrt(a/x)*x-%c];
 612 [method,ode_check(eqn,%[1])];
 613 [lie,0];
 614
 615 /* 97 */
 616 (pn_(97),ans:contrib_ode(eqn:x*'diff(y,x)^2=(a-x)^2,y,x));
 617 [-((3*y+sqrt(x)*(2*x-6*a))/3)=%c];
 618 solve(ans[1],y);
 619 [y=-((2*x^(3/2)-6*a*sqrt(x)+3*%c)/3)];
 620 [method,ode_check(eqn,%)];
 621 [lie,0];
 622
 623 /* 98 */
 624 (pn_(98),ans:contrib_ode(eqn:x*'diff(y,x)^2=y,y,x));
 625 [x^2*(y^2+(-2*x-2*%c)*y+x^2-2*%c*x+%c^2) = 0];
 626 ans:solve(expand(%[1]),y);
 627 [y = -2*sqrt(%c*x)+x+%c,y = 2*sqrt(%c*x)+x+%c];
 628 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2])];
 629 [lagrange,0,0];
 630
 631 /* 99 */
 632 (pn_(99),ans:contrib_ode(eqn:x*'diff(y,x)^2+x-2*y=0,y,x));
 633 [[x=%c*%e^-(2*(log(%t-1)-1/(%t-1))),y=(%t^2+1)*x/2]];
 634 [method,ode_check(eqn,ans[1])];
 635 [lagrange,0];
 636
 637 /* 100 */
 638 (pn_(100),ans:contrib_ode(eqn:x*'diff(y,x)^2+'diff(y,x)=y,y,x));
 639 [[x = (log(%t)-%t+%c)/(%t-1)^2,y = %t^2*x+%t]];
 640 [method,ode_check(eqn,ans[1])];
 641 [lagrange,0];
 642