Internal problem ID [7983]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 403.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {a \left (y^{\prime }\right )^{2}+y^{\prime } b -y=0} \end {gather*}
✓ Solution by Maple
Time used: 1.112 (sec). Leaf size: 247
dsolve(a*diff(y(x),x)^2+b*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {{\mathrm e}^{-\frac {b \ln \left (\frac {1}{4 a}\right )+2 b \LambertW \left (\frac {2 \,{\mathrm e}^{-\frac {c_{1}}{b}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{b}}}{b \sqrt {\frac {1}{a}}}\right )+2 c_{1}+2 b -2 x}{2 b}} \left ({\mathrm e}^{-\frac {b \ln \left (\frac {1}{4 a}\right )+2 b \LambertW \left (\frac {2 \,{\mathrm e}^{-\frac {c_{1}}{b}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{b}}}{b \sqrt {\frac {1}{a}}}\right )+2 c_{1}+2 b -2 x}{2 b}}+2 b \right )}{4 a} \\ y \relax (x ) = \frac {b^{2} \left (\LambertW \left (-\frac {2 \sqrt {a}\, {\mathrm e}^{-\frac {c_{1}}{b}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{b}}}{b}\right )+2\right ) \LambertW \left (-\frac {2 \sqrt {a}\, {\mathrm e}^{-\frac {c_{1}}{b}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{b}}}{b}\right )}{4 a} \\ y \relax (x ) = \frac {b^{2} \left (\LambertW \left (\frac {2 \sqrt {a}\, {\mathrm e}^{-\frac {c_{1}}{b}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{b}}}{b}\right )+2\right ) \LambertW \left (\frac {2 \sqrt {a}\, {\mathrm e}^{-\frac {c_{1}}{b}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{b}}}{b}\right )}{4 a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.787 (sec). Leaf size: 123
DSolve[-y[x] + b*y'[x] + a*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} a+b^2}+b \log \left (a \left (b-\sqrt {4 \text {$\#$1} a+b^2}\right )\right )}{2 a}\&\right ]\left [\frac {x}{2 a}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} a+b^2}-b \log \left (\sqrt {4 \text {$\#$1} a+b^2}+b\right )}{2 a}\&\right ]\left [-\frac {x}{2 a}+c_1\right ] \\ y(x)\to 0 \\ \end{align*}