Internal problem ID [7962]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 382.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+y^{\prime } a x -x^{2} b -c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 146
dsolve(diff(y(x),x)^2+a*x*diff(y(x),x)-b*x^2-c = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {x \sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}}{4}-\frac {c \ln \left (\sqrt {a^{2}+4 b}\, x +\sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}\right )}{\sqrt {a^{2}+4 b}}-\frac {x^{2} a}{4}+c_{1} \\ y \relax (x ) = \frac {x \sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}}{4}+\frac {c \ln \left (\sqrt {a^{2}+4 b}\, x +\sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}\right )}{\sqrt {a^{2}+4 b}}-\frac {x^{2} a}{4}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.195 (sec). Leaf size: 169
DSolve[-c - b*x^2 + a*x*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} x \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}-a x\right )-\frac {c \log \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}-x \sqrt {a^2+4 b}\right )}{\sqrt {a^2+4 b}}+c_1 \\ y(x)\to -\frac {1}{4} x \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}+a x\right )+\frac {c \log \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}-x \sqrt {a^2+4 b}\right )}{\sqrt {a^2+4 b}}+c_1 \\ \end{align*}