60.1.258 problem 259

Internal problem ID [10272]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 259
Date solved : Monday, January 27, 2025 at 06:42:50 PM
CAS classification : [_Bernoulli]

\begin{align*} 2 x^{2} y y^{\prime }-y^{2}-x^{2} {\mathrm e}^{x -\frac {1}{x}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 53

dsolve(2*x^2*y(x)*diff(y(x),x)-y(x)^2-x^2*exp(x-1/x)=0,y(x), singsol=all)
 

\begin{align*} y &= \sqrt {c_{1} {\mathrm e}^{-\frac {1}{x}}+{\mathrm e}^{\frac {\left (x -1\right ) \left (x +1\right )}{x}}} \\ y &= -\sqrt {c_{1} {\mathrm e}^{-\frac {1}{x}}+{\mathrm e}^{\frac {\left (x -1\right ) \left (x +1\right )}{x}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.935 (sec). Leaf size: 50

DSolve[2*x^2*y[x]*D[y[x],x]-y[x]^2-x^2*Exp[x-1/x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -e^{\left .-\frac {1}{2}\right /x} \sqrt {e^x+c_1} \\ y(x)\to e^{\left .-\frac {1}{2}\right /x} \sqrt {e^x+c_1} \\ \end{align*}