31.6.5 problem 5
Internal
problem
ID
[5754]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
5
Date
solved
:
Monday, January 27, 2025 at 01:12:49 PM
CAS
classification
:
[_quadrature]
\begin{align*} y&=a y^{\prime }+b {y^{\prime }}^{2} \end{align*}
✓ Solution by Maple
Time used: 1.302 (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: 0.922 (sec). Leaf size: 123
DSolve[y[x]==a*D[y[x],x]+b*(D[y[x],x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} b+a^2}+a \log \left (b \left (\sqrt {4 \text {$\#$1} b+a^2}-a\right )\right )}{2 b}\&\right ]\left [\frac {x}{2 b}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} b+a^2}-a \log \left (\sqrt {4 \text {$\#$1} b+a^2}+a\right )}{2 b}\&\right ]\left [-\frac {x}{2 b}+c_1\right ] \\
y(x)\to 0 \\
\end{align*}