29.33.16 problem 978

Internal problem ID [5555]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 978
Date solved : Monday, January 27, 2025 at 11:46:48 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{2}+2 y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.154 (sec). Leaf size: 107

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

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

✓ Solution by Mathematica

Time used: 8.189 (sec). Leaf size: 143

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

\begin{align*} y(x)\to -\frac {\sqrt {-2 x^2-4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-2 x^2-4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-2 x^2+4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-2 x^2+4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}} \\ \end{align*}