29.32.25 problem 960

Internal problem ID [5537]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 32
Problem number : 960
Date solved : Monday, January 27, 2025 at 11:35:51 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 8.050 (sec). Leaf size: 310

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

\begin{align*} y(x)\to c_1 \\ \text {Solve}\left [\frac {\left (2-\frac {2 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}\right ) \left (\frac {\frac {6 x^3}{x^2-a y(x)}-4 x}{\sqrt [3]{x^3}}+4\right ) \left (\left (1-\frac {x \left (2 a y(x)+x^2\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}\right ) \log \left (\frac {2-\frac {2 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}}{\sqrt [3]{2}}\right )+\left (\frac {2 a x y(x)+x^3}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}-1\right ) \log \left (\frac {\frac {\frac {6 x^3}{x^2-a y(x)}-4 x}{\sqrt [3]{x^3}}+4}{\sqrt [3]{2}}\right )-3\right )}{18 \sqrt [3]{2} \left (-\frac {\left (2 a y(x)+x^2\right )^3}{\left (x^2-a y(x)\right )^3}+\frac {3 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}-2\right )}&=\frac {2\ 2^{2/3} x \log (x)}{9 \sqrt [3]{x^3}}+c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}