29.24.20 problem 682

Internal problem ID [5273]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 682
Date solved : Tuesday, January 28, 2025 at 02:41:38 PM
CAS classification : [_rational]

\begin{align*} \left (a -3 x^{2}-y^{2}\right ) y y^{\prime }+x \left (a -x^{2}+y^{2}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 122

dsolve((a-3*x^2-y(x)^2)*y(x)*diff(y(x),x)+x*(a-x^2+y(x)^2) = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.365 (sec). Leaf size: 39

DSolve[(a-3*x^2-y[x]^2)*y[x]*D[y[x],x]+x*(a-x^2+y[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {1}{2} \left (\frac {a+2 y(x)^2}{x^2+y(x)^2}+\log \left (x^2+y(x)^2\right )\right )=c_1,y(x)\right ] \]