83.23.16 problem 16

Internal problem ID [19226]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 16
Date solved : Monday, March 31, 2025 at 06:57:55 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.170 (sec). Leaf size: 129

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

\begin{align*} y &= -4 x^{2} \\ y &= 0 \\ y &= \frac {\left (\sqrt {2}\, c_1 -2 x \right ) x \,c_1^{2}}{2 c_1^{2}-4 x^{2}} \\ y &= -\frac {\left (\sqrt {2}\, c_1 +2 x \right ) x \,c_1^{2}}{2 c_1^{2}-4 x^{2}} \\ y &= -\frac {2 \left (-c_1 x +\sqrt {2}\right ) x}{c_1 \left (c_1^{2} x^{2}-2\right )} \\ y &= \frac {2 \left (c_1 x +\sqrt {2}\right ) x}{c_1 \left (c_1^{2} x^{2}-2\right )} \\ \end{align*}

✓ Mathematica. Time used: 0.742 (sec). Leaf size: 79

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

\begin{align*} y(x)\to -\frac {x (\cosh (2 c_1)+\sinh (2 c_1))}{x+i \cosh (c_1)+i \sinh (c_1)} \\ y(x)\to \frac {x (\cosh (2 c_1)+\sinh (2 c_1))}{-x+i \cosh (c_1)+i \sinh (c_1)} \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

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

Timed Out