Internal problem ID [11235]
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)]`]]
\[ \boxed {x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}-x^{2} y^{2}=x^{4}} \]
✓ Solution by Maple
Time used: 0.578 (sec). Leaf size: 58
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 \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: 0.366 (sec). Leaf size: 60
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 \frac {1}{2} x e^{-x-c_1} \left (-1+e^{2 (x+c_1)}\right ) \\ y(x)\to \frac {1}{2} \left (x e^{-x+c_1}-x e^{x-c_1}\right ) \\ \end{align*}