Internal problem ID [10221]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
28. Summary. Page 59
Problem number: Ex 11.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }\right )^{2}-2 y x y^{\prime }+y^{2}-x^{2} y^{2}-x^{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 51
dsolve(x^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)^2=x^2*y(x)^2+x^4,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {x \left (\frac {{\mathrm e}^{2 x}}{c_{1}^{2}}-1\right ) {\mathrm e}^{-x} c_{1}}{2} \\ y \relax (x ) = \frac {x \left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{-x}}{2 c_{1}} \\ y \relax (x ) = x c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.312 (sec). Leaf size: 26
DSolve[x^2*(y'[x])^2-2*x*y[x]*y'[x]+y[x]^2==x^2*y[x]^2+x^4,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*}