8.5.24 problem 24

Internal problem ID [752]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 24
Date solved : Monday, January 27, 2025 at 03:02:46 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 47

dsolve(y(x)^3/exp(2*x)+2*x*diff(y(x),x) = 2*x*y(x),y(x), singsol=all)
 

\begin{align*} y &= \frac {\sqrt {\left (\ln \left (x \right )+c_1 \right ) {\mathrm e}^{2 x}}}{\ln \left (x \right )+c_1} \\ y &= \frac {\sqrt {\left (\ln \left (x \right )+c_1 \right ) {\mathrm e}^{2 x}}}{-\ln \left (x \right )-c_1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.333 (sec). Leaf size: 41

DSolve[y[x]^3/Exp[2*x]+2*x*D[y[x],x] == 2*x*y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {e^x}{\sqrt {\log (x)+c_1}} \\ y(x)\to \frac {e^x}{\sqrt {\log (x)+c_1}} \\ y(x)\to 0 \\ \end{align*}