9.5.15 problem 15

Internal problem ID [2951]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 9, page 38
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 06:11:41 AM
CAS classification : [[_homogeneous, `class D`], _Bernoulli]

\begin{align*} 2 x^{2} y y^{\prime }+x^{4} {\mathrm e}^{x}-2 x y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 7.049 (sec). Leaf size: 45

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

\begin{align*} y(x)&\to -\sqrt {-x^2 \left (e^x-c_1\right )}\\ y(x)&\to \sqrt {-x^2 \left (e^x-c_1\right )} \end{align*}

✓ Sympy. Time used: 0.277 (sec). Leaf size: 26

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

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