29.29.4 problem 826

Internal problem ID [5409]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 826
Date solved : Sunday, March 30, 2025 at 08:10:35 AM
CAS classification : [_separable]

\begin{align*} {y^{\prime }}^{2}-x y \left (x^{2}+y^{2}\right ) y^{\prime }+x^{4} y^{4}&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 38

ode:=diff(y(x),x)^2-x*y(x)*(x^2+y(x)^2)*diff(y(x),x)+x^4*y(x)^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 \,{\mathrm e}^{\frac {x^{4}}{4}} \\ y &= \frac {1}{\sqrt {-x^{2}+c_1}} \\ y &= -\frac {1}{\sqrt {-x^{2}+c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.202 (sec). Leaf size: 60

ode=(D[y[x],x])^2-x*y[x]*(x^2+y[x]^2)*D[y[x],x]+x^4*y[x]^4==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {1}{\sqrt {-x^2-2 c_1}} \\ y(x)\to \frac {1}{\sqrt {-x^2-2 c_1}} \\ y(x)\to c_1 e^{\frac {x^4}{4}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.115 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*y(x)**4 - x*(x**2 + y(x)**2)*y(x)*Derivative(y(x), x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \sqrt {- \frac {1}{C_{1} + x^{2}}}, \ y{\left (x \right )} = \sqrt {- \frac {1}{C_{1} + x^{2}}}, \ y{\left (x \right )} = C_{1} e^{\frac {x^{4}}{4}}\right ] \]