Internal problem ID [8058]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 478.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (a y+b \right ) \left (\left (y^{\prime }\right )^{2}+1\right )-c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.256 (sec). Leaf size: 88
dsolve((a*y(x)+b)*(diff(y(x),x)^2+1)-c = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {-b +c}{a} \\ x -\left (\int _{}^{y \relax (x )}\frac {a \textit {\_a} +b}{\sqrt {-\left (a \textit {\_a} +b \right ) \left (a \textit {\_a} +b -c \right )}}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}-\frac {a \textit {\_a} +b}{\sqrt {-\left (a \textit {\_a} +b \right ) \left (a \textit {\_a} +b -c \right )}}d \textit {\_a} \right )-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.48 (sec). Leaf size: 154
DSolve[-c + (b + a*y[x])*(1 + y'[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {c \text {ArcTan}\left (\frac {\sqrt {-\text {$\#$1} a-b+c}}{\sqrt {\text {$\#$1} a+b}}\right )+\sqrt {\text {$\#$1} a+b} \sqrt {-\text {$\#$1} a-b+c}}{a}\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [-\frac {c \text {ArcTan}\left (\frac {\sqrt {-\text {$\#$1} a-b+c}}{\sqrt {\text {$\#$1} a+b}}\right )+\sqrt {\text {$\#$1} a+b} \sqrt {-\text {$\#$1} a-b+c}}{a}\&\right ][x+c_1] \\ y(x)\to \frac {c-b}{a} \\ \end{align*}