60.1.487 problem 490

Internal problem ID [10501]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 490
Date solved : Monday, January 27, 2025 at 08:22:50 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 y^{2}-x^{2}+a&=0 \end{align*}

✓ Solution by Maple

Time used: 0.105 (sec). Leaf size: 83

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

\begin{align*} y &= -\frac {\sqrt {4 x^{2}-2 a}}{2} \\ y &= \frac {\sqrt {4 x^{2}-2 a}}{2} \\ y &= -\frac {\sqrt {-8 c_{1}^{2}+16 c_{1} x -4 x^{2}-2 a}}{2} \\ y &= \frac {\sqrt {-8 c_{1}^{2}+16 c_{1} x -4 x^{2}-2 a}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.694 (sec). Leaf size: 63

DSolve[a - x^2 + 2*y[x]^2 - 2*x*y[x]*D[y[x],x] + y[x]^2*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {-\frac {a}{2}-x^2+4 c_1 x-2 c_1{}^2} \\ y(x)\to \sqrt {-\frac {a}{2}-x^2+4 c_1 x-2 c_1{}^2} \\ \end{align*}