share/contrib/diffequations/tests/rtestode_murphy_2_4.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Murphy equations 2.301 - 2.300
   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 /* 313 */
  24 (pn_(313),ans:contrib_ode(eqn:x*'diff(y,x)^3-y*'diff(y,x)^2+a=0,y,x));
  25 [y=(%c^3*x+a)/%c^2,y =(3*sqrt(3)*%i-3)*a^(1/3)*x^(2/3)/(2*4^(1/3)),y=-((3*sqrt(3)*%i+3)*a^(1/3)*x^(2/3)/(2*4^(1/3))),y=3*a^(1/3)*x^(2/3)/4^(1/3)];
  26 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3]),ode_check(eqn,ans[4])];
  27 [clairaut,0,0,0,0];
  28
  29 /* 314 */
  30 (pn_(314),ans:contrib_ode(eqn:x*'diff(y,x)^3+(2*x-y)*'diff(y,x)^2+(x-2*y)*'diff(y,x)+1-y=0,y,x));
  31 [y=((%c^3+2*%c^2+%c)*x+1)/(%c^2+2*%c+1),y=3*(-sqrt(3)*%i/2-1/2)*x^(2/3)/4^(1/3)-x,y=3*(sqrt(3)*%i/2-1/2)*x^(2/3)/4^(1/3)-x,y=(9*x^(2/3)/4^(1/3)-3*x)/3];
  32 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3]),ode_check(eqn,ans[4])];
  33 [clairaut,0,0,0,0];
  34
  35 /* 320 */
  36 (pn_(320),ans:contrib_ode(eqn:x^2*'diff(y,x)^3-2*x*y*'diff(y,x)^2+y^2*'diff(y,x)+1=0,y,x),0);
  37 0$
  38 [method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3]),ode_check(eqn,ans[4])];
  39 [clairaut,0,0,0,0];
  40
  41 /* 367 FIXME - check second solution */
  42 (pn_(367),ans:contrib_ode(eqn:'diff(y,x)^n+x*'diff(y,x)-y=0,y,x));
  43 [y=%c*x+%c^n,[-y+%t*x+%t^n=0,(%t*x+n*%t^n)/%t=0]];
  44 [method,ode_check(eqn,ans[1])];
  45 [clairaut,0];
  46
  47 /* 373 */
  48 (pn_(373),ans:contrib_ode(eqn:2*sqrt(a*'diff(y,x))+x*'diff(y,x)-y=0,y,x));
  49 [y=%c*x+2*sqrt(%c*a),[-y+%t*x+2*sqrt(a*%t)=0,(sqrt(a*%t)*x+a)/sqrt(a*%t)=0]];
  50 assume(a>0,x<0);
  51 [a>0,x<0];
  52 first(solve(ans[2][2],%t));
  53 %t=a/x^2;
  54 ans[2][1],%;
  55 -y-a/x=0;
  56 s:first(solve(%,y));
  57 y=-a/x;
  58 [method,ode_check(eqn,ans[1]),ode_check(eqn,s)];
  59 [clairaut,0,0];
  60 forget(a>0,x<0);
  61 [a>0,x<0];
  62
  63 /* 375 */
  64 (pn_(375),ans:contrib_ode(eqn:2*(1+'diff(y,x))^(3/2)+3*x*'diff(y,x)-3*y=0,y,x));
  65 [y=(3*%c*x+sqrt(%c+1)*(2*%c+2))/3,[-3*y+3*%t*x+2*(%t+1)^(3/2)=0,3*x+3*sqrt(%t+1)=0]];
  66 assume(u>0);
  67 [u>0];
  68 s:subst(u^2-1,%t,ans[2]);
  69 [-3*y+3*(u^2-1)*x+2*u^3=0,3*x+3*u=0];
  70 solve(s[2],u);
  71 [u=-x]; /* so x<0 for singular solution */
  72 s[1],%[1];
  73 -3*y-2*x^3+3*x*(x^2-1)=0;
  74 ans[2]:first(solve(%,y));
  75 y=(x^3-3*x)/3;
  76 [method,ode_check(eqn,ans[1]),(assume(x<0),ode_check(eqn,ans[2]))];
  77 [clairaut,0,0];
  78 forget(u>0,x<0);
  79 [u>0,x<0];
  80
  81 /* 381 FIXME - check second solution */
  82 (pn_(381),ans:contrib_ode(eqn:sqrt(a^2+b^2*'diff(y,x)^2)+x*'diff(y,x)-y=0,y,x));
  83 [y=%c*x+sqrt(%c^2*b^2+a^2),[-y+%t*x+sqrt(b^2*%t^2+a^2)=0,(sqrt(b^2*%t^2+a^2)*x+b^2*%t)/sqrt(b^2*%t^2+a^2)=0]];
  84 [method,ode_check(eqn,ans[1])];
  85 [clairaut,0];
  86
  87 /* 382 FIXME - check second solution */
  88 (pn_(382),ans:contrib_ode(eqn:sqrt(a+2*b*'diff(y,x)+c*'diff(y,x)^2)+(x*'diff(y,x)-y)*sqrt(a*c-b^2)=0,y,x));
  89 [y=(%c*sqrt(a*c-b^2)*x+sqrt(%c^2*c+2*%c*b+a))/sqrt(a*c-b^2),
  90   [sqrt(a*c-b^2)*(%t*x-y)+sqrt(%t^2*c+2*%t*b+a)=0,
  91    (sqrt(a*c-b^2)*sqrt(c*%t^2+2*b*%t+a)*x+c*%t+b)/sqrt(c*%t^2+2*b*%t+a)=0]];
  92 [method,ode_check(eqn,ans[1])];
  93 [clairaut,0];
  94
  95 /* 388 FIXME - check second solution */
  96 (pn_(388),ans:contrib_ode(eqn:a*(1+'diff(y,x))^(1/3)+x*'diff(y,x)-y=0,y,x));
  97 [y=%c*x+(%c+1)^(1/3)*a,[-y+%t*x+a*(%t+1)^(1/3)=0,(3*(%t+1)^(2/3)*x+a)/(3*(%t+1)^(2/3))=0]];
  98 [method,ode_check(eqn,ans[1])];
  99 [clairaut,0];
 100
 101 /* 389 FIXME - check second solution */
 102 (pn_(389),ans:contrib_ode(eqn:(y-x*'diff(y,x))*sqrt(1+'diff(y,x)^2)+a*'diff(y,x)=0,y,x));
 103 [y=(%c*sqrt(%c^2+1)*x-%c*a)/sqrt(%c^2+1),[sqrt(%t^2+1)*(y-%t*x)+%t*a=0,(%t*y-2*%t^2*x-x+sqrt(%t^2+1)*a)/sqrt(%t^2+1) = 0]];
 104 [method,ode_check(eqn,ans[1])];
 105 [clairaut,0];
 106
 107 /* 390 */
 108 (pn_(390),ans:contrib_ode(eqn:y=x*'diff(y,x)+cos('diff(y,x)),y,x));
 109 [y=cos(%c)+%c*x, [y-%t*x-cos(%t)=0,sin(%t)-x=0]];
 110 first(solve(ans[2][2],%t));
 111 %t=asin(x);
 112 s:first(solve(ev(ans[2][1],%),y));
 113 y=x*asin(x)+sqrt(1-x^2);
 114 [method,ode_check(eqn,ans[1]),ode_check(eqn,s)];
 115 [clairaut,0,0];
 116
 117 /* 396  FIXME - check second solution */
 118 (pn_(396),ans:contrib_ode(eqn:(1+'diff(y,x)^2)*sin(y-x*'diff(y,x))^2=1,y,x));
 119 [y=%c*x+asin(1/sqrt(%c^2+1)),[(%t^2+1)*sin(y-%t*x)^2-1=0,2*%t*sin(y-%t*x)^2-(2*%t^2*x+2*x)*cos(y-%t*x)*sin(y-%t*x)=0]];
 120 [method,ode_check(eqn,ans[1])];
 121 [clairaut,0];
 122
 123 /* 397 FIXME - check second solution */
 124 (pn_(397),ans:contrib_ode(eqn:'diff(y,x)*(acos('diff(y,x))-x)+sqrt(1-'diff(y,x)^2)+y=0,y,x));
 125 [y=%c*x-%c*acos(%c)-sqrt(1-%c^2),
 126   [y+%t*(acos(%t)-x)+sqrt(1-%t^2)=0,
 127   (sqrt(1-%t^2)*(acos(%t)-x)-2*%t)/sqrt(1-%t^2)=0]];
 128 [method,ode_check(eqn,ans[1])];
 129 [clairaut,0];