Internal problem ID [3902]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 7
Problem number: 5.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {y-a y^{\prime }-b \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 247
dsolve(y(x)=a*diff(y(x),x)+b*(diff(y(x),x))^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {{\mathrm e}^{-\frac {a \ln \left (\frac {1}{4 b}\right )+2 a \LambertW \left (\frac {2 \,{\mathrm e}^{-\frac {c_{1}}{a}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{a}}}{a \sqrt {\frac {1}{b}}}\right )+2 c_{1}+2 a -2 x}{2 a}} \left ({\mathrm e}^{-\frac {a \ln \left (\frac {1}{4 b}\right )+2 a \LambertW \left (\frac {2 \,{\mathrm e}^{-\frac {c_{1}}{a}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{a}}}{a \sqrt {\frac {1}{b}}}\right )+2 c_{1}+2 a -2 x}{2 a}}+2 a \right )}{4 b} \\ y \relax (x ) = \frac {a^{2} \left (\LambertW \left (-\frac {2 \sqrt {b}\, {\mathrm e}^{-\frac {c_{1}}{a}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{a}}}{a}\right )+2\right ) \LambertW \left (-\frac {2 \sqrt {b}\, {\mathrm e}^{-\frac {c_{1}}{a}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{a}}}{a}\right )}{4 b} \\ y \relax (x ) = \frac {a^{2} \left (\LambertW \left (\frac {2 \sqrt {b}\, {\mathrm e}^{-\frac {c_{1}}{a}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{a}}}{a}\right )+2\right ) \LambertW \left (\frac {2 \sqrt {b}\, {\mathrm e}^{-\frac {c_{1}}{a}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{a}}}{a}\right )}{4 b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.773 (sec). Leaf size: 123
DSolve[y[x]==a*y'[x]+b*(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} b+a^2}+a \log \left (b \left (a-\sqrt {4 \text {$\#$1} b+a^2}\right )\right )}{2 b}\&\right ]\left [\frac {x}{2 b}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} b+a^2}-a \log \left (\sqrt {4 \text {$\#$1} b+a^2}+a\right )}{2 b}\&\right ]\left [-\frac {x}{2 b}+c_1\right ] \\ y(x)\to 0 \\ \end{align*}