share/contrib/diffequations/tests/rtestode_murphy_1_4.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* ODE tests - Murphy equations 1.301 - 1.400
   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 /* 301 Riccati */
  20
  21 /* 304 - Abel eqn of 1st kind */
  22 (pn_(304),ans:contrib_ode(eqn:(1+x^2)*'diff(y,x)=1+y^2-2*x*y*(1+y^2),y,x));
  23 false;
  24
  25 /* 311 */
  26 (pn_(311),ans:contrib_ode(eqn:(a^2+x^2)*'diff(y,x)=a^2+3*x*y-2*y^2,y,x));
  27  [y = (2*x^3*sqrt(x^2+a^2)+%c*x^4+x^2*(%c*(x^2+a^2)+%c*a^2)
  28                                 +%c*a^2*(x^2+a^2)+2*a^2*x*sqrt(x^2+a^2))
  29           /(2*%c*x*(x^2+a^2)+2*x^2*sqrt(x^2+a^2)+2*a^2*sqrt(x^2+a^2))];
  30 [method,ode_check(eqn,ans[1])];
  31 [riccati,0];
  32
  33 /* 324 Riccati */
  34
  35 /* 329 Riccati */
  36
  37 /* 333 Riccati
  38       Solution should simplify to y=x*(%c*x+sin(x))/(%c*x-sin(x)) ?
  39 */
  40 /*  FIXME:  This test now prompts - 2006-10-02
  41 (pn_(333),ans:contrib_ode(eqn:2*x^2*'diff(y,x)=2*x*y+(1-x*cot(x))*(x^2-y^2),y,x));
  42 [y=sqrt(4*x^3*cot(x)-4*x^2)*sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)*(%c*sin('integrate(sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)/sqrt(4*x^3*cot(x)-4*x^2),x))-cos('integrate(sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)/sqrt(4*x^3*cot(x)-4*x^2),x)))/((2*x^2*cot(x)^2-4*x*cot(x)+2)*sin('integrate(sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)/sqrt(4*x^3*cot(x)-4*x^2),x))+(2*%c*x^2*cot(x)^2-4*%c*x*cot(x)+2*%c)*cos('integrate(sqrt(-x^3*cot(x)^3+3*x^2*cot(x)^2-3*x*cot(x)+1)/sqrt(4*x^3*cot(x)-4*x^2),x)))];
  43 [method,ode_check(eqn,ans[1])];
  44 [riccati,0];
  45 */
  46
  47 /* 336 Riccati */
  48 (pn_(336),ans:contrib_ode(eqn:x*(1-2*x)*'diff(y,x)=4*x-(1+4*x)*y+y^2,y,x));
  49 [-((x*y-2*x^2)/(y-1)) = %c];
  50 ans:first(solve(ans[1],y));
  51 y=(2*x^2+%c)/(x+%c);
  52 [exact,ode_check(eqn,ans)];
  53 [exact,0];
  54
  55 /* 337  - check this*/
  56 (pn_(337),ans:contrib_ode(eqn:2*x*(1-x)*'diff(y,x)+x+(1-2*x)*y=0,y,x));
  57 [y = %e^-(log(2*x^2-2*x)/2)*(log(2*sqrt(2*x^2-2*x)/sqrt(2)+2*x-1)
  58          /(2*sqrt(2))+sqrt(2*x^2-2*x)/2+%c)];
  59 [method,radcan(ode_check(eqn,ans[1]))];
  60 [linear,0];
  61
  62 /* 338 - Runs "forever" with odelin 2006-12-10 */
  63 /* (pn_(338),ans:contrib_ode(eqn:2*x*(1-x)*'diff(y,x)+x+(1-2*x)*y^2=0,y,x));
  64 [[y='diff(%u,x,1)*(2*x^2-2*x)/(%u*(2*x-1)),-'diff(%u,x,1)*((2*x-1)*(4*x-2)/(2*x^2-2*x)^2-2/(2*x^2-2*x))-'diff(%u,x,2)*(2*x-1)/(2*x^2-2*x)+%u*(2*x-1)^2/((2*x-2)*(2*x^2-2*x)^2)=0]];
  65 method;
  66 riccati; */
  67
  68 /* 345 - Abel eqn of 1st kind */
  69 (pn_(345),ans:contrib_ode(eqn:(a+b*x)^2*'diff(y,x)+c*y^2+(a+b*x)*y^3=0,y,x));
  70 false;
  71
  72 /* 347 */
  73 (pn_(347),ans:contrib_ode(eqn:x^3*'diff(y,x)=3-x^2+x^2*y,y,x));
  74 [y=x*((x^2-1)/x^3+%c)];
  75 [method,ode_check(eqn,ans[1])];
  76 [linear,0];
  77
  78 /* 348 Riccati */
  79 (pn_(348),ans:contrib_ode(eqn:x^3*'diff(y,x)=x^4+y^2,y,x));
  80 [x=%c*%e^-(x^2/(y-x^2))];
  81 ans:first(solve(ans[1],y));
  82 y=(log(%c/x)+1)*x^2/log(%c/x);
  83 [method,ode_check(eqn,ans)];
  84 [genhom,0];
  85
  86 /* 352 Riccati */
  87 (pn_(352),ans:contrib_ode(eqn:x^3*'diff(y,x)+20+x^2*y*(1-x^2*y)=0,y,x));
  88 [x=%c*%e^-((log(x^2*y+5)-log(x^2*y-4))/9)];
  89 ans:first(solve((ans[1]/%c)^9,y));
  90 y=-((5*x^9+4*%c^9)/(x^11-%c^9*x^2));
  91 [method,ode_check(eqn,ans)];
  92 [genhom,0];
  93
  94 /* 353 Riccati */
  95 (pn_(353),ans:contrib_ode(eqn:x^3*'diff(y,x)+3+(3-2*x)*x^2*y-x^6*y^2=0,y,x));
  96 [y=-((3*%e^(4*x)-%c)/(x^3*%e^(4*x)+%c*x^3))];
  97 [method,ode_check(eqn,ans[1])];
  98 [riccati,0];
  99
 100 /* 366 Riccati */
 101
 102 /* 371 Riccati */
 103
 104 /* 373 */
 105 assume(a>0);
 106 [a>0];
 107 ans:contrib_ode(eqn:x^4*'diff(y,x)+a^2+x^4*y^2=0,y,x);
 108 [y=(x*tan((%c*x-a)/x)+a)/(x^2*tan((%c*x-a)/x))];
 109 [method,ode_check(eqn,ans[1])];
 110 [riccati,0];
 111 forget(a>0);
 112 [a>0];
 113
 114 /* 377 */
 115 (pn_(377),ans:contrib_ode(eqn:x*(1-x^3)*'diff(y,x)=x^2+(1-2*x*y)*y,y,x));
 116 [y = (2*%c*(x-1)*x^6+x*(3*%c*(x-1)*(x^2+x+1)-2*(x-1))
 117                            +x^4*(2*(x-1)-3*%c*(x-1)*(x^2+x+1))
 118                            +x^5*(%c*(x^2+x+1)+%c*(x-1))
 119                            +x^2*(-%c*(x^2+x+1)-%c*(x-1))
 120                            +x^3*(x^2+x+(1-2*%c)*(x-1)+1)-x^2-2*x)
 121           /(3*%c*(x-1)*x^2*(x^2+x+1)+3*(x-1)*(x^2+x+1))];
 122 [method,ode_check(eqn,ans[1])];
 123 [riccati,0];
 124
 125 /* 380 Riccati */
 126
 127 /* 383 Abel eqn of 1st kind */
 128 (pn_(383),ans:contrib_ode(eqn:x^7*'diff(y,x)+5*x^3*y^2+2*(1+x^2)*y^3=0,y,x));
 129 false;
 130
 131 /* 385 Riccati */
 132 (pn_(385),ans:contrib_ode(eqn:x^n*'diff(y,x)=x^(2*n-1)-y^2,y,x));
 133 [y = -(((%c*'bessel_y(n,2*sqrt(-x))+'bessel_j(n,2*sqrt(-x))
 134                                   -%c*'bessel_y(n-2,2*sqrt(-x))
 135                                   -'bessel_j(n-2,2*sqrt(-x)))
 136    *sqrt(-x)
 137    +n*(%c*'bessel_y(n-1,2*sqrt(-x))+'bessel_j(n-1,2*sqrt(-x)))
 138    -%c*'bessel_y(n-1,2*sqrt(-x))-'bessel_j(n-1,2*sqrt(-x)))
 139    *x^(n-1)
 140    /(2*%c*'bessel_y(n-1,2*sqrt(-x))+2*'bessel_j(n-1,2*sqrt(-x))))];
 141 [method,ode_check(eqn,ans[1])];
 142 [riccati,0];
 143
 144 /* 387 Riccati */
 145 (pn_(387),ans:contrib_ode(eqn:x^n*'diff(y,x)+x^(2*n-2)+y^2+(1-n)*x^(n-1)*y=0,y,x));
 146 [x=%c*%e^-atan(x^(1-n)*y)];
 147 ans:map(log,ans[1]/%c);
 148 log(x/%c)=-atan(x^(1-n)*y);
 149 ans:map(tan,ans);
 150 tan(log(x/%c))=-x^(1-n)*y;
 151 ans:first(solve(ans,y));
 152 y=-x^(n-1)*tan(log(x/%c));
 153 [method,ode_check(eqn,%)];
 154 [genhom,0];
 155
 156 /* 388 Riccati */
 157