23.2.81 problem 83
Internal
problem
ID
[5436]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
83
Date
solved
:
Tuesday, September 30, 2025 at 12:43:19 PM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} {y^{\prime }}^{2}&={\mathrm e}^{4 x -2 y} \left (y^{\prime }-1\right ) \end{align*}
✓ Maple. Time used: 0.608 (sec). Leaf size: 253
ode:=diff(y(x),x)^2 = exp(4*x-2*y(x))*(diff(y(x),x)-1);
dsolve(ode,y(x), singsol=all);
\begin{align*}
x -\frac {\sqrt {{\mathrm e}^{8 x -4 y}-4 \,{\mathrm e}^{4 x -2 y}}\, {\mathrm e}^{-4 x +2 y} \operatorname {arctanh}\left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y}+1}}\right )}{2 \sqrt {-4 \,{\mathrm e}^{-4 x +2 y}+1}}+\frac {\ln \left ({\mathrm e}^{-2 x +y}\right )}{2}-\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y}+1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y}-1\right )}{4}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y}-1\right )}{4}-c_1 &= 0 \\
x +\frac {\ln \left ({\mathrm e}^{-2 x +y}\right )}{2}-\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y}+1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y}-1\right )}{4}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y}-1\right )}{4}+\frac {\sqrt {{\mathrm e}^{8 x -4 y}-4 \,{\mathrm e}^{4 x -2 y}}\, {\mathrm e}^{-4 x +2 y} \operatorname {arctanh}\left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y}+1}}\right )}{2 \sqrt {-4 \,{\mathrm e}^{-4 x +2 y}+1}}-c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.644 (sec). Leaf size: 197
ode=(D[y[x],x])^2==Exp[4 x -2 y[x]] (D[y[x],x]-1);
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((1 - Derivative(y(x), x))*exp(4*x - 2*y(x)) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out