1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
|
(load("contrib_ode"),0);
0$
/* ODE tests - Murphy equations 2.301 - 2.300
Reference:
G M Murphy, Ordinary Differential Equations and Their
Solutions, Van Nostrand, 1960, pp 15-23
First Order and Second or Higher Degree, p278 ff
*/
/* Don't kill(all); It messes up trigsimp */
/* Eliminate parameter t from singular solution */
(elim_(ans,x,y,t):=block([s],s:first(solve(ans[2],t)),s:solve(ev(ans[1],s),y),s[1]),done);
done;
/* Print ode number*/
(pn_(n_):=print("Murphy ODE 2.",n_),true);
true;
/* 313 */
(pn_(313),ans:contrib_ode(eqn:x*'diff(y,x)^3-y*'diff(y,x)^2+a=0,y,x));
[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)];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3]),ode_check(eqn,ans[4])];
[clairault,0,0,0,0];
/* 314 */
(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));
[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];
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3]),ode_check(eqn,ans[4])];
[clairault,0,0,0,0];
/* 320 */
(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);
0$
[method,ode_check(eqn,ans[1]),ode_check(eqn,ans[2]),ode_check(eqn,ans[3]),ode_check(eqn,ans[4])];
[clairault,0,0,0,0];
/* 367 FIXME - check second solution */
(pn_(367),ans:contrib_ode(eqn:'diff(y,x)^n+x*'diff(y,x)-y=0,y,x));
[y=%c*x+%c^n,[-y+%t*x+%t^n=0,(%t*x+n*%t^n)/%t=0]];
[method,ode_check(eqn,ans[1])];
[clairault,0];
/* 373 */
(pn_(373),ans:contrib_ode(eqn:2*sqrt(a*'diff(y,x))+x*'diff(y,x)-y=0,y,x));
[y=%c*x+2*sqrt(%c*a),[-y+%t*x+2*sqrt(a*%t)=0,(sqrt(a*%t)*x+a)/sqrt(a*%t)=0]];
assume(a>0,x<0);
[a>0,x<0];
first(solve(ans[2][2],%t));
%t=a/x^2;
ans[2][1],%;
-y-a/x=0;
s:first(solve(%,y));
y=-a/x;
[method,ode_check(eqn,ans[1]),ode_check(eqn,s)];
[clairault,0,0];
forget(a>0,x<0);
[a>0,x<0];
/* 375 */
(pn_(375),ans:contrib_ode(eqn:2*(1+'diff(y,x))^(3/2)+3*x*'diff(y,x)-3*y=0,y,x));
[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]];
assume(u>0);
[u>0];
s:subst(u^2-1,%t,ans[2]);
[-3*y+3*(u^2-1)*x+2*u^3=0,3*x+3*u=0];
solve(s[2],u);
[u=-x]; /* so x<0 for singular solution */
s[1],%[1];
-3*y-2*x^3+3*x*(x^2-1)=0;
ans[2]:first(solve(%,y));
y=(x^3-3*x)/3;
[method,ode_check(eqn,ans[1]),(assume(x<0),ode_check(eqn,ans[2]))];
[clairault,0,0];
forget(u>0,x<0);
[u>0,x<0];
/* 381 FIXME - check second solution */
(pn_(381),ans:contrib_ode(eqn:sqrt(a^2+b^2*'diff(y,x)^2)+x*'diff(y,x)-y=0,y,x));
[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]];
[method,ode_check(eqn,ans[1])];
[clairault,0];
/* 382 FIXME - check second solution */
(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));
[y=(%c*sqrt(a*c-b^2)*x+sqrt(%c^2*c+2*%c*b+a))/sqrt(a*c-b^2),
[sqrt(a*c-b^2)*(%t*x-y)+sqrt(%t^2*c+2*%t*b+a)=0,
(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]];
[method,ode_check(eqn,ans[1])];
[clairault,0];
/* 388 FIXME - check second solution */
(pn_(388),ans:contrib_ode(eqn:a*(1+'diff(y,x))^(1/3)+x*'diff(y,x)-y=0,y,x));
[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]];
[method,ode_check(eqn,ans[1])];
[clairault,0];
/* 389 FIXME - check second solution */
(pn_(389),ans:contrib_ode(eqn:(y-x*'diff(y,x))*sqrt(1+'diff(y,x)^2)+a*'diff(y,x)=0,y,x));
[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]];
[method,ode_check(eqn,ans[1])];
[clairault,0];
/* 390 */
(pn_(390),ans:contrib_ode(eqn:y=x*'diff(y,x)+cos('diff(y,x)),y,x));
[y=cos(%c)+%c*x, [y-%t*x-cos(%t)=0,sin(%t)-x=0]];
first(solve(ans[2][2],%t));
%t=asin(x);
s:first(solve(ev(ans[2][1],%),y));
y=x*asin(x)+sqrt(1-x^2);
[method,ode_check(eqn,ans[1]),ode_check(eqn,s)];
[clairault,0,0];
/* 396 FIXME - check second solution */
(pn_(396),ans:contrib_ode(eqn:(1+'diff(y,x)^2)*sin(y-x*'diff(y,x))^2=1,y,x));
[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]];
[method,ode_check(eqn,ans[1])];
[clairault,0];
/* 397 FIXME - check second solution */
(pn_(397),ans:contrib_ode(eqn:'diff(y,x)*(acos('diff(y,x))-x)+sqrt(1-'diff(y,x)^2)+y=0,y,x));
[y=%c*x-%c*acos(%c)-sqrt(1-%c^2),
[y+%t*(acos(%t)-x)+sqrt(1-%t^2)=0,
(sqrt(1-%t^2)*(acos(%t)-x)-2*%t)/sqrt(1-%t^2)=0]];
[method,ode_check(eqn,ans[1])];
[clairault,0];
|
