62.17.11 problem Ex 11

Internal problem ID [12911]
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
Date solved : Tuesday, January 28, 2025 at 04:43:10 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 }+y^{2}&=x^{2} y^{2}+x^{4} \end{align*}

✓ Solution by Maple

Time used: 0.998 (sec). Leaf size: 56

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

✓ Solution by Mathematica

Time used: 0.155 (sec). Leaf size: 26

DSolve[x^2*(D[y[x],x])^2-2*x*y[x]*D[y[x],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*}