share/contrib/diffequations/tests/rtestode_murphy_1_3.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Murphy equations 1.201 - 1.300
   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 /* 201 */
  20 assume(b>0);
  21 [b>0];
  22 (pn_(201),ans:contrib_ode(eqn:x*'diff(y,x)=y+a*sqrt(y^2+b*x^2),y,x));
  23 [x=%c*%e^(x*asinh(y/(sqrt(b)*x))/(a*abs(x)))];
  24 assume(x>0);
  25 [x>0];
  26 ratsimp(ans[1]);
  27 x = %c*%e^(asinh(y/(sqrt(b)*x))/a);
  28 map(log,%/%c);
  29 log(x/%c) = asinh(y/(sqrt(b)*x))/a;
  30 map(sinh,a*%);
  31 sinh(a*log(x/%c)) = y/(sqrt(b)*x);
  32 solve(%,y);
  33 [y = sqrt(b)*x*sinh(a*log(x/%c))];
  34 [method,ode_check(eqn,%[1])];
  35 [genhom,0];
  36 forget(b>0,x>0);
  37 [b>0,x>0];
  38
  39 /* 240 riccati */
  40
  41 /* 249 */
  42 ans:contrib_ode(eqn:3*x*'diff(y,x)=3*x^(2/3)+(1-3*y)*y,y,x);
  43 [y=tanh(3*x^(1/3)+%c)*x^(1/3)];
  44 [method,ode_check(eqn,ans[1])];
  45 [riccati,0];
  46
  47 /* 266 */
  48 (pn_(266),ans:contrib_ode(eqn:x^2*'diff(y,x)=a+b*x^n+x^2*y^2,y,x));
  49 [[y=-'diff(%u,x,1)/%u,%u*(b*x^(n-2)+a/x^2)+'diff(%u,x,2)=0]];
  50 method;
  51 riccati;
  52
  53 /* 267 Riccati */
  54 (pn_(267),ans:contrib_ode(eqn:x^2*'diff(y,x)+2+x*y*(4+x*y)=0,y,x));
  55 [x=%c*(x*y+2)/(x*y+1)];
  56 method;
  57 genhom;
  58
  59 /* 268 */
  60 (pn_(268),ans:contrib_ode(eqn:x^2*'diff(y,x)+2+a*x*(1-x*y)-x^2*y^2=0,y,x));
  61  [y = -((%c*a^3*x^3*%e^(a*x)-%c*a^2*x^2*%e^(a*x)+2*%c*a*x*%e^(a*x)
  62                                  -2*%c*%e^(a*x)-1)
  63           /(x*(2*%c*%e^(a*x)+1)+%c*a^2*x^3*%e^(a*x)-2*%c*a*x^2*%e^(a*x)))];
  64 [method,ode_check(eqn,ans[1])];
  65 [riccati,0];
  66
  67 /* 270 */
  68 (pn_(270),ans:contrib_ode(eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2,y,x));
  69 [[y=-'diff(%u,x,1)/(%u*c),%u*c^2*(b*x^(n-2)+a/x^2)+'diff(%u,x,2)*c=0]];
  70 method;
  71 riccati;
  72
  73 /* 271 Riccati */
  74 (pn_(271),assume(4*a*c-b^2-2*b-1>0),ans:contrib_ode(eqn:x^2*'diff(y,x)=a+b*x*y+c*x^2*y^2,y,x),forget(4*a*c-b^2-2*b-1>0),ans);
  75 [x=%c*%e^(2*atan((2*c*x*y+b+1)/sqrt(4*a*c-b^2-2*b-1))/sqrt(4*a*c-b^2-2*b-1))];
  76 method;
  77 genhom;
  78
  79 /* 272 */
  80 (pn_(272),ans:contrib_ode(eqn:x^2*'diff(y,x)=a+b*x*y+c*x^4*y^2,y,x));
  81 [y = (sqrt(a)*(%c*bessel_y(-((b+5)/2),-sqrt(a)*sqrt(c)*x)
  82                     +bessel_j(-((b+5)/2),-sqrt(a)*sqrt(c)*x)
  83                     -%c*bessel_y(-((b+1)/2),-sqrt(a)*sqrt(c)*x)
  84                     -bessel_j(-((b+1)/2),-sqrt(a)*sqrt(c)*x))*sqrt(c)*x
  85          +b*(-%c*bessel_y(-((b+3)/2),-sqrt(a)*sqrt(c)*x)
  86             -bessel_j(-((b+3)/2),-sqrt(a)*sqrt(c)*x))
  87          -3*%c*bessel_y(-((b+3)/2),-sqrt(a)*sqrt(c)*x)
  88          -3*bessel_j(-((b+3)/2),-sqrt(a)*sqrt(c)*x))
  89          /((2*%c*bessel_y(-((b+3)/2),-sqrt(a)*sqrt(c)*x)
  90           +2*bessel_j(-((b+3)/2),-sqrt(a)*sqrt(c)*x))
  91           *c*x^3)];
  92 [method,ode_check(eqn,ans[1])];
  93 [riccati,0];
  94
  95 /* 275 - Abel eqn of 1st kind */
  96 (pn_(275),ans:contrib_ode(eqn:x^2*'diff(y,x)=a*x^2*y^2-a*y^3,y,x));
  97 false;
  98
  99 /* 276 - Abel eqn of 1st kind */
 100 (pn_(276),ans:contrib_ode(eqn:x^2*'diff(y,x)+a*y^2+b*x^2*y^3=0,y,x));
 101 false;
 102
 103 /* 300 Riccati */
 104 (pn_(300),ans:contrib_ode(eqn:(1-x^2)*'diff(y,x)=1-(2*x-y)*y,y,x),0);
 105 0$
 106 [method,ode_check(eqn,ans[1])];
 107 [riccati,0];