15.7.2 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 : Monday, January 27, 2025 at 07:05:34 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 112

dsolve(y(x)^3*diff(y(x),x)+x*y(x)^4=x*exp(-x^2),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*}

✓ Solution by Mathematica

Time used: 0.414 (sec). Leaf size: 200

DSolve[y[x]^3*D[y[x],x]+x*y[x]^4==x*exp[-x^2],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*}