9.7.2 problem 2

Internal problem ID [2983]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 11, page 45
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 06:13:46 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{3} y^{\prime }+x y^{4}&=x \,{\mathrm e}^{-x^{2}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 112

ode:=y(x)^3*diff(y(x),x)+x*y(x)^4 = x*exp(-x^2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= {\mathrm e}^{-x^{2}} {\left (\left (2 \,{\mathrm e}^{x^{2}}+c_1 \right ) {\mathrm e}^{2 x^{2}}\right )}^{{1}/{4}} \\ y &= -{\mathrm e}^{-x^{2}} {\left (\left (2 \,{\mathrm e}^{x^{2}}+c_1 \right ) {\mathrm e}^{2 x^{2}}\right )}^{{1}/{4}} \\ y &= -i {\mathrm e}^{-x^{2}} {\left (\left (2 \,{\mathrm e}^{x^{2}}+c_1 \right ) {\mathrm e}^{2 x^{2}}\right )}^{{1}/{4}} \\ y &= i {\mathrm e}^{-x^{2}} {\left (\left (2 \,{\mathrm e}^{x^{2}}+c_1 \right ) {\mathrm e}^{2 x^{2}}\right )}^{{1}/{4}} \\ \end{align*}

✓ Mathematica. Time used: 0.236 (sec). Leaf size: 200

ode=y[x]^3*D[y[x],x]+x*y[x]^4==x*exp[-x^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -e^{-\frac {x^2}{2}} \sqrt [4]{4 \int _1^xe^{2 K[1]^2} \exp \left (-K[1]^2\right ) K[1]dK[1]+c_1}\\ y(x)&\to -i e^{-\frac {x^2}{2}} \sqrt [4]{4 \int _1^xe^{2 K[1]^2} \exp \left (-K[1]^2\right ) K[1]dK[1]+c_1}\\ y(x)&\to i e^{-\frac {x^2}{2}} \sqrt [4]{4 \int _1^xe^{2 K[1]^2} \exp \left (-K[1]^2\right ) K[1]dK[1]+c_1}\\ y(x)&\to e^{-\frac {x^2}{2}} \sqrt [4]{4 \int _1^xe^{2 K[1]^2} \exp \left (-K[1]^2\right ) K[1]dK[1]+c_1} \end{align*}

✓ Sympy. Time used: 1.862 (sec). Leaf size: 82

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)**4 - x*exp(-x**2) + y(x)**3*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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