15.19.7 problem 7

Internal problem ID [3291]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 7
Date solved : Sunday, March 30, 2025 at 01:32:48 AM
CAS classification : [_separable]

\begin{align*} y^{2} {y^{\prime }}^{2}+x y y^{\prime }-2 x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 49

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

\begin{align*} y &= \sqrt {x^{2}+c_1} \\ y &= -\sqrt {x^{2}+c_1} \\ y &= \sqrt {-2 x^{2}+c_1} \\ y &= -\sqrt {-2 x^{2}+c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 80

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

\begin{align*} y(x)\to -\sqrt {2} \sqrt {-x^2+c_1} \\ y(x)\to \sqrt {2} \sqrt {-x^2+c_1} \\ y(x)\to -\sqrt {x^2+2 c_1} \\ y(x)\to \sqrt {x^2+2 c_1} \\ \end{align*}

✓ Sympy. Time used: 1.019 (sec). Leaf size: 48

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

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