82.17.4 problem Ex. 4
Internal
problem
ID
[18820]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
III.
Equations
of
the
first
order
but
not
of
the
first
degree.
Problems
at
page
37
Problem
number
:
Ex.
4
Date
solved
:
Tuesday, January 28, 2025 at 12:21:31 PM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} {\mathrm e}^{4 x} \left (y^{\prime }-1\right )+{\mathrm e}^{2 y} {y^{\prime }}^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 1.104 (sec). Leaf size: 95
dsolve(exp(4*x)*(diff(y(x),x)-1)+exp(2*y(x))*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\operatorname {arctanh}\left (\operatorname {RootOf}\left (-1+\left ({\mathrm e}^{8}+4 \,{\mathrm e}^{\operatorname {RootOf}\left (-4 \,{\mathrm e}^{\textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}-2 x +2 c_{1} +4\right )^{2}+{\mathrm e}^{8}\right )}\right ) \textit {\_Z}^{2}\right ) {\mathrm e}^{4}\right )+2 c_{1} \\
y \left (x \right ) &= \operatorname {arctanh}\left (\operatorname {RootOf}\left (-1+\left ({\mathrm e}^{8}+4 \,{\mathrm e}^{\operatorname {RootOf}\left (-4 \,{\mathrm e}^{\textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}-2 x +2 c_{1} +4\right )^{2}+{\mathrm e}^{8}\right )}\right ) \textit {\_Z}^{2}\right ) {\mathrm e}^{4}\right )+2 c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.958 (sec). Leaf size: 197
DSolve[Exp[4*x]*(D[y[x],x]-1)+Exp[2*y[x]]*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {y(x)}{2}-\frac {e^{-2 x} \sqrt {4 e^{2 y(x)+4 x}+e^{8 x}} \text {arctanh}\left (\frac {e^{2 x}}{\sqrt {4 e^{2 y(x)}+e^{4 x}}}\right )}{2 \sqrt {4 e^{2 y(x)}+e^{4 x}}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {e^{-2 x} \sqrt {4 e^{2 y(x)+4 x}+e^{8 x}} \text {arctanh}\left (\frac {e^{2 x}}{\sqrt {4 e^{2 y(x)}+e^{4 x}}}\right )}{2 \sqrt {4 e^{2 y(x)}+e^{4 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*}