29.29.17 problem 839
Internal
problem
ID
[5422]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
29
Problem
number
:
839
Date
solved
:
Monday, January 27, 2025 at 11:21:54 AM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} 4 {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x -2 y} y^{\prime }-{\mathrm e}^{2 x -2 y}&=0 \end{align*}
✓ Solution by Maple
Time used: 3.502 (sec). Leaf size: 115
dsolve(4*diff(y(x),x)^2+2*exp(2*x-2*y(x))*diff(y(x),x)-exp(2*x-2*y(x)) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= c_{1} -\operatorname {arctanh}\left (\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}} \cosh \left (-\frac {\textit {\_Z}}{2}-x +c_{1} \right )^{2}+16 \,{\mathrm e}^{2 \textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}-x +c_{1} \right )^{2}-8 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right ) \\
y \left (x \right ) &= c_{1} +\operatorname {arctanh}\left (\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}} \cosh \left (-\frac {\textit {\_Z}}{2}-x +c_{1} \right )^{2}+16 \,{\mathrm e}^{2 \textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}-x +c_{1} \right )^{2}-8 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.985 (sec). Leaf size: 176
DSolve[4 (D[y[x],x])^2+2 Exp[2 x-2 y[x]] D[y[x],x]-Exp[2 x-2 y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [y(x)-\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \text {arctanh}\left (\frac {e^x}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \text {arctanh}\left (\frac {e^x}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}+y(x)&=c_1,y(x)\right ] \\
y(x)\to \frac {1}{2} \left (\log \left (-\frac {e^{4 x}}{4}\right )-2 x\right ) \\
\end{align*}