29.33.17 problem 979

Internal problem ID [5556]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 979
Date solved : Monday, January 27, 2025 at 11:46:51 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.122 (sec). Leaf size: 83

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

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {4 x^{2}-2 a}}{2} \\ y \left (x \right ) &= \frac {\sqrt {4 x^{2}-2 a}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {-8 c_{1}^{2}+16 c_{1} x -4 x^{2}-2 a}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-8 c_{1}^{2}+16 c_{1} x -4 x^{2}-2 a}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.678 (sec). Leaf size: 63

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

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