Internal problem ID [8237]
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)]`]]
\[ \boxed {y^{\prime }+\frac {a x}{2}+\frac {b}{2}-x \sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c}=0} \]
✓ Solution by Maple
Time used: 0.297 (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 \left (x \right ) = -\frac {a^{2} x^{2}+2 a b x +b^{2}-4 c}{4 a} \\ \frac {a \,x^{2}}{\left (-a \,x^{2}+\sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y \left (x \right )-4 c}\right ) \left (-a^{2} x^{4}+a^{2} x^{2}+2 a b x +4 a y \left (x \right )+b^{2}-4 c \right )}+\frac {\sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y \left (x \right )-4 c}}{\left (-a \,x^{2}+\sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y \left (x \right )-4 c}\right ) \left (-a^{2} x^{4}+a^{2} x^{2}+2 a b x +4 a y \left (x \right )+b^{2}-4 c \right )}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 42.349 (sec). Leaf size: 50
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 (-e^{-\frac {a \left (x^2-2 c_1\right )}{b}}\right )-(a x+b)^2+4 c}{4 a} \\ \end{align*}