Internal problem ID [8056]
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]
\[ \boxed {\left (a y+b \right ) \left ({y^{\prime }}^{2}+1\right )-c=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 393
dsolve((a*y(x)+b)*(diff(y(x),x)^2+1)-c = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {c -b}{a} \\ x -\frac {b \arctan \left (\frac {\sqrt {a^{2}}\, \left (y \left (x \right )-\frac {-b a -a \left (-c +b \right )}{2 a^{2}}\right )}{\sqrt {-a^{2} y \left (x \right )^{2}+\left (-b a -a \left (-c +b \right )\right ) y \left (x \right )-b \left (-c +b \right )}}\right )}{\sqrt {a^{2}}}-a \left (-\frac {\sqrt {-a^{2} y \left (x \right )^{2}+\left (-2 b a +a c \right ) y \left (x \right )-b \left (-c +b \right )}}{a^{2}}+\frac {\left (-2 b a +a c \right ) \arctan \left (\frac {\sqrt {a^{2}}\, \left (y \left (x \right )-\frac {-2 b a +a c}{2 a^{2}}\right )}{\sqrt {-a^{2} y \left (x \right )^{2}+\left (-2 b a +a c \right ) y \left (x \right )-b \left (-c +b \right )}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )-c_{1} = 0 \\ x +\frac {b \arctan \left (\frac {\sqrt {a^{2}}\, \left (y \left (x \right )-\frac {-b a -a \left (-c +b \right )}{2 a^{2}}\right )}{\sqrt {-a^{2} y \left (x \right )^{2}+\left (-b a -a \left (-c +b \right )\right ) y \left (x \right )-b \left (-c +b \right )}}\right )}{\sqrt {a^{2}}}+a \left (-\frac {\sqrt {-a^{2} y \left (x \right )^{2}+\left (-2 b a +a c \right ) y \left (x \right )-b \left (-c +b \right )}}{a^{2}}+\frac {\left (-2 b a +a c \right ) \arctan \left (\frac {\sqrt {a^{2}}\, \left (y \left (x \right )-\frac {-2 b a +a c}{2 a^{2}}\right )}{\sqrt {-a^{2} y \left (x \right )^{2}+\left (-2 b a +a c \right ) y \left (x \right )-b \left (-c +b \right )}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.43 (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*}