29.29.8 problem 830
Internal
problem
ID
[5413]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
29
Problem
number
:
830
Date
solved
:
Monday, January 27, 2025 at 11:21:31 AM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} {y^{\prime }}^{2}&={\mathrm e}^{4 x -2 y} \left (y^{\prime }-1\right ) \end{align*}
✓ Solution by Maple
Time used: 2.048 (sec). Leaf size: 307
dsolve(diff(y(x),x)^2 = exp(4*x-2*y(x))*(diff(y(x),x)-1),y(x), singsol=all)
\begin{align*}
\frac {-\frac {\sqrt {-4 \,{\mathrm e}^{-4 y \left (x \right )+8 x} {\mathrm e}^{-4 x +2 y \left (x \right )}+{\mathrm e}^{-4 y \left (x \right )+8 x}}\, {\mathrm e}^{-4 x +2 y \left (x \right )} \operatorname {arctanh}\left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}\right )}{2}+\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}\, \left (x -\frac {\ln \left (2 \,{\mathrm e}^{y \left (x \right )-2 x}+1\right )}{4}-c_{1} +\frac {\ln \left ({\mathrm e}^{y \left (x \right )-2 x}\right )}{2}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{y \left (x \right )-2 x}-1\right )}{4}\right )}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}} &= 0 \\
\frac {\frac {\sqrt {-4 \,{\mathrm e}^{-4 y \left (x \right )+8 x} {\mathrm e}^{-4 x +2 y \left (x \right )}+{\mathrm e}^{-4 y \left (x \right )+8 x}}\, {\mathrm e}^{-4 x +2 y \left (x \right )} \operatorname {arctanh}\left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}\right )}{2}+\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}\, \left (x -\frac {\ln \left (2 \,{\mathrm e}^{y \left (x \right )-2 x}+1\right )}{4}-c_{1} +\frac {\ln \left ({\mathrm e}^{y \left (x \right )-2 x}\right )}{2}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{y \left (x \right )-2 x}-1\right )}{4}\right )}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.110 (sec). Leaf size: 197
DSolve[(D[y[x],x])^2==Exp[4 x -2 y[x]] (D[y[x],x]-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {y(x)}{2}-\frac {e^{-2 x} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}} \text {arctanh}\left (\frac {e^{2 x}}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}\right )}{2 \sqrt {e^{4 x}-4 e^{2 y(x)}}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {e^{-2 x} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}} \text {arctanh}\left (\frac {e^{2 x}}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}\right )}{2 \sqrt {e^{4 x}-4 e^{2 y(x)}}}+\frac {y(x)}{2}&=c_1,y(x)\right ] \\
y(x)\to \frac {1}{2} \left (\log \left (\frac {e^{8 x}}{4}\right )-4 x\right ) \\
\end{align*}