82.8.20 problem 36-21
Internal
problem
ID
[21891]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
II.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
36.
Nonlinear
differential
equations.
Page
1203
Problem
number
:
36-21
Date
solved
:
Thursday, October 02, 2025 at 08:05:47 PM
CAS
classification
:
[_quadrature]
\begin{align*} 2 {y^{\prime }}^{2}-2 y y^{\prime }-1&=0 \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 69
ode:=2*diff(y(x),x)^2-2*y(x)*diff(y(x),x)-1 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
x +\frac {y^{2}}{2}+\frac {y \sqrt {y^{2}+2}}{2}+\operatorname {arcsinh}\left (\frac {\sqrt {2}\, y}{2}\right )-c_1 &= 0 \\
x +\frac {y^{2}}{2}-\frac {y \sqrt {y^{2}+2}}{2}-\operatorname {arcsinh}\left (\frac {\sqrt {2}\, y}{2}\right )-c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.342 (sec). Leaf size: 109
ode=D[y[x],x]^2-2*y[x]*D[y[x],x]-1+D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{4} \left (\text {$\#$1} \left (\sqrt {\text {$\#$1}^2+2}-\text {$\#$1}\right )-2 \log \left (\sqrt {\text {$\#$1}^2+2}-\text {$\#$1}\right )\right )\&\right ]\left [\frac {x}{2}+c_1\right ]\\ y(x)&\to \text {InverseFunction}\left [\frac {1}{4} \left (\text {$\#$1} \left (\sqrt {\text {$\#$1}^2+2}+\text {$\#$1}\right )-2 \log \left (\sqrt {\text {$\#$1}^2+2}-\text {$\#$1}\right )\right )\&\right ]\left [-\frac {x}{2}+c_1\right ] \end{align*}
✓ Sympy. Time used: 0.589 (sec). Leaf size: 167
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*y(x)*Derivative(y(x), x) + 2*Derivative(y(x), x)**2 - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ x - \frac {\sqrt {y^{2}{\left (x \right )} + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} y{\left (x \right )}}{2} \right )}}{- \sqrt {y^{2}{\left (x \right )} + 2} + y{\left (x \right )}} + \frac {y{\left (x \right )} \operatorname {asinh}{\left (\frac {\sqrt {2} y{\left (x \right )}}{2} \right )}}{- \sqrt {y^{2}{\left (x \right )} + 2} + y{\left (x \right )}} - \frac {y{\left (x \right )}}{- \sqrt {y^{2}{\left (x \right )} + 2} + y{\left (x \right )}} = C_{1}, \ x - \frac {\sqrt {y^{2}{\left (x \right )} + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} y{\left (x \right )}}{2} \right )}}{\sqrt {y^{2}{\left (x \right )} + 2} + y{\left (x \right )}} - \frac {y{\left (x \right )} \operatorname {asinh}{\left (\frac {\sqrt {2} y{\left (x \right )}}{2} \right )}}{\sqrt {y^{2}{\left (x \right )} + 2} + y{\left (x \right )}} - \frac {y{\left (x \right )}}{\sqrt {y^{2}{\left (x \right )} + 2} + y{\left (x \right )}} = C_{1}\right ]
\]