29.31.9 problem 908

Internal problem ID [5488]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 908
Date solved : Monday, January 27, 2025 at 11:28:45 AM
CAS classification : [_separable]

\begin{align*} x^{2} {y^{\prime }}^{2}-4 x \left (y+2\right ) y^{\prime }+4 \left (y+2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 137

dsolve(x^2*diff(y(x),x)^2-4*x*(2+y(x))*diff(y(x),x)+4*(2+y(x))*y(x) = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -2 \\ y \left (x \right ) &= \frac {-2 \sqrt {2}\, \sqrt {c_{1} x^{2}}+x^{2}}{c_{1}} \\ y \left (x \right ) &= \frac {2 \sqrt {2}\, \sqrt {c_{1} x^{2}}+x^{2}}{c_{1}} \\ y \left (x \right ) &= \frac {\left (-8 c_{1}^{2}+x^{2}\right ) \left (-2 \sqrt {2}\, c_{1} +x \right ) x}{\left (-4 c_{1} \sqrt {2}\, x +8 c_{1}^{2}+x^{2}\right ) c_{1}^{2}} \\ y \left (x \right ) &= \frac {\left (-8 c_{1}^{2}+x^{2}\right ) \left (2 \sqrt {2}\, c_{1} +x \right ) x}{\left (4 c_{1} \sqrt {2}\, x +8 c_{1}^{2}+x^{2}\right ) c_{1}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.198 (sec). Leaf size: 69

DSolve[x^2 (D[y[x],x])^2-4 x(2+y[x])D[y[x],x]+4(2+y[x])y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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