Internal problem ID [8239]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 659.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {a x}{2}+\frac {b}{2}-x \sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c}=0} \end {gather*}
✓ Solution by Maple
Time used: 1.296 (sec). Leaf size: 204
dsolve(diff(y(x),x) = -1/2*a*x-1/2*b+x*(a^2*x^2+2*a*b*x+b^2+4*a*y(x)-4*c)^(1/2),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {x^{2} a^{2}+2 a b x +b^{2}-4 c}{4 a} \\ \frac {x^{2} a}{\left (-x^{2} a +\sqrt {x^{2} a^{2}+2 a b x +b^{2}+4 a y \relax (x )-4 c}\right ) \left (-a^{2} x^{4}+x^{2} a^{2}+2 a b x +4 a y \relax (x )+b^{2}-4 c \right )}+\frac {\sqrt {x^{2} a^{2}+2 a b x +b^{2}+4 a y \relax (x )-4 c}}{\left (-x^{2} a +\sqrt {x^{2} a^{2}+2 a b x +b^{2}+4 a y \relax (x )-4 c}\right ) \left (-a^{2} x^{4}+x^{2} a^{2}+2 a b x +4 a y \relax (x )+b^{2}-4 c \right )}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.761 (sec). Leaf size: 53
DSolve[y'[x] == -1/2*b - (a*x)/2 + x*Sqrt[b^2 - 4*c + 2*a*b*x + a^2*x^2 + 4*a*y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b^2 \log ^2\left (-\frac {e^{-\frac {2 a \left (x^2-2 c_1\right )}{b}}}{a}\right )-4 (a x+b)^2+16 c}{16 a} \\ \end{align*}