Internal problem ID [8029]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 451.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
\[ \boxed {\left (x^{2}+a \right ) {y^{\prime }}^{2}-2 x y^{\prime } y+y^{2}+b=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 78
dsolve((x^2+a)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)^2+b = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\sqrt {-a b \left (x^{2}+a \right )}}{a} \\ y \left (x \right ) = -\frac {\sqrt {-a b \left (x^{2}+a \right )}}{a} \\ y \left (x \right ) = x c_{1} -\sqrt {-a \,c_{1}^{2}-b} \\ y \left (x \right ) = x c_{1} +\sqrt {-a \,c_{1}^{2}-b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.316 (sec). Leaf size: 96
DSolve[b + y[x]^2 - 2*x*y[x]*y'[x] + (a + x^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x-\sqrt {-b-a c_1{}^2} \\ y(x)\to \sqrt {-b-a c_1{}^2}+c_1 x \\ y(x)\to -\frac {\sqrt {-b \left (a+x^2\right )}}{\sqrt {a}} \\ y(x)\to \frac {\sqrt {-b \left (a+x^2\right )}}{\sqrt {a}} \\ \end{align*}