Internal problem ID [4211]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 978.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 y^{2}=x^{2}} \]
✓ Solution by Maple
Time used: 0.047 (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: 7.875 (sec). Leaf size: 233
DSolve[y[x]^2 (y'[x])^2-2 x y[x] y'[x]-x^2+2 y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x^2-4 \sqrt {2} x \cosh (c_1)-4 \sqrt {2} x \sinh (c_1)-4 \cosh (2 c_1)-4 \sinh (2 c_1)} \\ y(x)\to \sqrt {-x^2-4 \sqrt {2} x \cosh (c_1)-4 \sqrt {2} x \sinh (c_1)-4 \cosh (2 c_1)-4 \sinh (2 c_1)} \\ y(x)\to -\sqrt {-x^2+4 \sqrt {2} x \cosh (c_1)+4 \sqrt {2} x \sinh (c_1)-4 \cosh (2 c_1)-4 \sinh (2 c_1)} \\ y(x)\to \sqrt {-x^2+4 \sqrt {2} x \cosh (c_1)+4 \sqrt {2} x \sinh (c_1)-4 \cosh (2 c_1)-4 \sinh (2 c_1)} \\ y(x)\to -\sqrt {-x^2} \\ y(x)\to \sqrt {-x^2} \\ \end{align*}