29.37.6 problem 1119
Internal
problem
ID
[5664]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
37
Problem
number
:
1119
Date
solved
:
Monday, January 27, 2025 at 01:02:21 PM
CAS
classification
:
[_quadrature]
\begin{align*} \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }&=y \end{align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 112
dsolve(sqrt(1+diff(y(x),x)^2)+a*diff(y(x),x) = y(x),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: 0.996 (sec). Leaf size: 210
DSolve[Sqrt[1+(D[y[x],x])^2]+ a*D[y[x],x]==y[x],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*}