29.30.39 problem 899

Internal problem ID [5479]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 30
Problem number : 899
Date solved : Monday, January 27, 2025 at 11:26:59 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{4}+\left (-x^{2}+1\right ) y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.520 (sec). Leaf size: 56

dsolve(x^2*diff(y(x),x)^2-2*x*diff(y(x),x)*y(x)-x^4+(-x^2+1)*y(x)^2 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= -\frac {x \left ({\mathrm e}^{x}-c_{1}^{2} {\mathrm e}^{-x}\right )}{2 c_{1}} \\ y \left (x \right ) &= \frac {x \left (c_{1}^{2} {\mathrm e}^{x}-{\mathrm e}^{-x}\right )}{2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.150 (sec). Leaf size: 26

DSolve[x^2 (D[y[x],x])^2-2 x y[x] D[y[x],x]-x^4+(1-x^2)y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -x \sinh (x-c_1) \\ y(x)\to x \sinh (x+c_1) \\ \end{align*}