29.27.8 problem 774
Internal
problem
ID
[5358]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
27
Problem
number
:
774
Date
solved
:
Sunday, March 30, 2025 at 08:02:54 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2}-2 y^{\prime }-y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.029 (sec). Leaf size: 66
ode:=diff(y(x),x)^2-2*diff(y(x),x)-y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {-\sqrt {1+y^{2}}+\operatorname {arcsinh}\left (y\right ) y-1+\left (x -c_1 \right ) y}{y} &= 0 \\
\frac {\sqrt {1+y^{2}}-\operatorname {arcsinh}\left (y\right ) y-1+\left (x -c_1 \right ) y}{y} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.964 (sec). Leaf size: 104
ode=(D[y[x],x])^2-2*D[y[x],x]-y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {$\#$1}^2+1}+\text {$\#$1} \log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )+1}{\text {$\#$1}}\&\right ][-x+c_1] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {$\#$1}^2+1}}{\text {$\#$1}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )+\frac {1}{\text {$\#$1}}\&\right ][x+c_1] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 2.181 (sec). Leaf size: 80
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x)**2 + Derivative(y(x), x)**2 - 2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} - \frac {\sqrt {y^{2}{\left (x \right )} + 1}}{y{\left (x \right )}} + \operatorname {asinh}{\left (y{\left (x \right )} \right )} - \frac {1}{y{\left (x \right )}} & \text {for}\: \left |{y^{2}{\left (x \right )}}\right | > 1 \\\frac {\sqrt {y^{2}{\left (x \right )} + 1}}{y{\left (x \right )}} - \operatorname {asinh}{\left (y{\left (x \right )} \right )} - \frac {1}{y{\left (x \right )}} & \text {otherwise} \end {cases} = C_{1} - x, \ x + \frac {\sqrt {y^{2}{\left (x \right )} + 1}}{y{\left (x \right )}} - \operatorname {asinh}{\left (y{\left (x \right )} \right )} - \frac {1}{y{\left (x \right )}} = C_{1}\right ]
\]