Internal problem ID [8813]
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]
\[ \boxed {\left (a y+b \right ) \left (1+{y^{\prime }}^{2}\right )=c} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 162
dsolve((a*y(x)+b)*(diff(y(x),x)^2+1)-c = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {-b +c}{a} \\ \frac {-\arctan \left (\frac {2 a y \left (x \right )+2 b -c}{2 \sqrt {-\left (a y \left (x \right )+b \right ) \left (a y \left (x \right )+b -c \right )}}\right ) c +2 \sqrt {-\left (a y \left (x \right )+b \right ) \left (a y \left (x \right )+b -c \right )}+\left (2 x -2 c_{1} \right ) a}{2 a} &= 0 \\ \frac {\arctan \left (\frac {2 a y \left (x \right )+2 b -c}{2 \sqrt {-\left (a y \left (x \right )+b \right ) \left (a y \left (x \right )+b -c \right )}}\right ) c -2 \sqrt {-\left (a y \left (x \right )+b \right ) \left (a y \left (x \right )+b -c \right )}+\left (2 x -2 c_{1} \right ) a}{2 a} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.409 (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 \arctan \left (\frac {\sqrt {\text {$\#$1} a+b}}{\sqrt {-\text {$\#$1} a-b+c}}\right )-\sqrt {\text {$\#$1} a+b} \sqrt {-\text {$\#$1} a-b+c}}{a}\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [\frac {c \arctan \left (\frac {\sqrt {\text {$\#$1} a+b}}{\sqrt {-\text {$\#$1} a-b+c}}\right )-\sqrt {\text {$\#$1} a+b} \sqrt {-\text {$\#$1} a-b+c}}{a}\&\right ][x+c_1] \\ y(x)\to \frac {c-b}{a} \\ \end{align*}