Internal problem ID [7645]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 64.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\sqrt {\frac {a y^{2}+b y+c}{a \,x^{2}+x b +c}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 124
dsolve(diff(y(x),x) - sqrt((a*y(x)^2+b*y(x)+c)/(a*x^2+b*x+c))=0,y(x), singsol=all)
\[ -\frac {\sqrt {\frac {a y \relax (x )^{2}+b y \relax (x )+c}{a \,x^{2}+b x +c}}\, \sqrt {a \,x^{2}+b x +c}\, \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right )}{\sqrt {a y \relax (x )^{2}+b y \relax (x )+c}\, \sqrt {a}}+\frac {\ln \left (\frac {a y \relax (x )+\frac {b}{2}}{\sqrt {a}}+\sqrt {a y \relax (x )^{2}+b y \relax (x )+c}\right )}{\sqrt {a}}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 6.42 (sec). Leaf size: 113
DSolve[y'[x]- Sqrt[(a*y[x]^2+b*y[x]+c)/(a*x^2+b*x+c)]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \sqrt {a} \sinh \left (\sqrt {a} c_1\right ) \sqrt {x (a x+b)+c}+(2 a x+b) \cosh \left (\sqrt {a} c_1\right )-b}{2 a} \\ y(x)\to -\frac {\sqrt {b^2-4 a c}+b}{2 a} \\ y(x)\to \frac {\sqrt {b^2-4 a c}-b}{2 a} \\ \end{align*}