Internal problem ID [4410]
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]
\[ \boxed {y-a y^{\prime }-b {y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 207
dsolve(y(x)=a*diff(y(x),x)+b*(diff(y(x),x))^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= {\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a \sqrt {\frac {1}{b}}}\right )-a +x -c_{1}}{a}} \left (a \sqrt {\frac {1}{b}}+{\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a \sqrt {\frac {1}{b}}}\right )-a +x -c_{1}}{a}}\right ) \\ y \left (x \right ) &= \frac {a^{2} \left (\operatorname {LambertW}\left (-\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (-\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a}\right )}{4 b} \\ y \left (x \right ) &= \frac {a^{2} \left (\operatorname {LambertW}\left (\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a}\right )}{4 b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.803 (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*}