29.37.10 problem 1126
Internal
problem
ID
[5668]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
37
Problem
number
:
1126
Date
solved
:
Tuesday, January 28, 2025 at 02:41:57 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.131 (sec). Leaf size: 338
dsolve(a*x*sqrt(1+diff(y(x),x)^2)+x*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*}
\frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a +y \left (x \right )}{x \left (a^{2}-1\right )}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\
\frac {x \sqrt {\frac {-2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+\left (a^{2}+1\right ) y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a -y \left (x \right )}{x \left (a^{2}-1\right )}\right )}{a}} c_{1}}{\sqrt {\frac {-2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+\left (a^{2}+1\right ) y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.700 (sec). Leaf size: 223
DSolve[a*x*Sqrt[1+(D[y[x],x])^2]+x*D[y[x],x] -y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\
\text {Solve}\left [\frac {-2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\
\end{align*}