31.6.7 problem 7
Internal
problem
ID
[5756]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
7
Date
solved
:
Monday, January 27, 2025 at 01:12:50 PM
CAS
classification
:
[_quadrature]
\begin{align*} y&=\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } \end{align*}
✓ Solution by Maple
Time used: 0.069 (sec). Leaf size: 112
dsolve(y(x)=a*diff(y(x),x)+sqrt(1+(diff(y(x),x))^2),y(x), singsol=all)
\begin{align*}
\left (\int _{}^{y \left (x \right )}\frac {1}{-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}-\int _{}^{y \left (x \right )}\frac {1}{-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_{1} +x &= 0 \\
-\left (\int _{}^{y \left (x \right )}\frac {1}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}+\int _{}^{y \left (x \right )}\frac {1}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_{1} +x &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.878 (sec). Leaf size: 210
DSolve[y[x]==a*D[y[x],x]+Sqrt[1+(D[y[x],x])^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {a \left (\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a+1\right )+\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}+a-1\right )\right )-(a+1) \log \left ((a-1) \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}\right )\right )}{a^2-1}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {a \left (\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a-1\right )+\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}+a+1\right )\right )-(a-1) \log \left ((a+1) \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}\right )\right )}{a^2-1}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\
y(x)\to 1 \\
\end{align*}