47.2.3 problem 3

Internal problem ID [7419]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 3
Date solved : Monday, January 27, 2025 at 02:53:47 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 2 x y^{\prime }&=y \left (2 x^{2}-y^{2}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 83

dsolve(2*x*diff(y(x),x)=y(x)*(2*x^2-y(x)^2),y(x), singsol=all)
 

\begin{align*} y &= \frac {\sqrt {2}\, \sqrt {\left (2 c_{1} -\operatorname {Ei}_{1}\left (-x^{2}\right )\right ) {\mathrm e}^{x^{2}}}}{-2 c_{1} +\operatorname {Ei}_{1}\left (-x^{2}\right )} \\ y &= \frac {\sqrt {2}\, \sqrt {\left (2 c_{1} -\operatorname {Ei}_{1}\left (-x^{2}\right )\right ) {\mathrm e}^{x^{2}}}}{2 c_{1} -\operatorname {Ei}_{1}\left (-x^{2}\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.284 (sec). Leaf size: 65

DSolve[2*x*D[y[x],x]==y[x]*(2*x^2-y[x]^2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {e^{\frac {x^2}{2}}}{\sqrt {\frac {\operatorname {ExpIntegralEi}\left (x^2\right )}{2}+c_1}} \\ y(x)\to \frac {e^{\frac {x^2}{2}}}{\sqrt {\frac {\operatorname {ExpIntegralEi}\left (x^2\right )}{2}+c_1}} \\ y(x)\to 0 \\ \end{align*}