Internal problem ID [7961]
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]
\[ \boxed {{y^{\prime }}^{2}+a x y^{\prime }-x^{2} b -c=0} \]
✓ Solution by Maple
Time used: 0.047 (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 \left (x \right ) = -\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 {a \,x^{2}}{4}+c_{1} \\ y \left (x \right ) = \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 {a \,x^{2}}{4}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.496 (sec). Leaf size: 194
DSolve[-c - b*x^2 + a*x*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 c \arctan \left (\frac {x \sqrt {-a^2-4 b}}{\sqrt {x^2 \left (a^2+4 b\right )+4 c}-2 \sqrt {c}}\right )}{\sqrt {-a^2-4 b}}-\frac {1}{4} x \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}+a x\right )+c_1 \\ y(x)\to \frac {2 c \arctan \left (\frac {x \sqrt {-a^2-4 b}}{\sqrt {x^2 \left (a^2+4 b\right )+4 c}-2 \sqrt {c}}\right )}{\sqrt {-a^2-4 b}}+\frac {1}{4} x \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}-a x\right )+c_1 \\ \end{align*}