81.7.6 problem 7
Internal
problem
ID
[18624]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
VI.
Certain
particular
forms
of
equations.
Exercises
at
page
74
Problem
number
:
7
Date
solved
:
Thursday, March 13, 2025 at 12:26:45 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}&=1 \end{align*}
✓ Maple. Time used: 0.025 (sec). Leaf size: 55
ode:=y(x)*diff(diff(y(x),x),x)-diff(y(x),x)^2 = 1;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}+{\mathrm e}^{\frac {-c_{2} -x}{c_{1}}}\right )}{2} \\
y \left (x \right ) &= \frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}+{\mathrm e}^{\frac {-c_{2} -x}{c_{1}}}\right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.931 (sec). Leaf size: 464
ode=y[x]*D[y[x],{x,2}]-(D[y[x],x])^2==1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 (-c_1)}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 (-c_1)}}\right )}{\sqrt {-e^{2 (-c_1)}} \sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 (-c_1)}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 (-c_1)}}\right )}{\sqrt {-e^{2 (-c_1)}} \sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2 - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(y(x)*Derivative(y(x), (x, 2)) - 1) + Derivative(y(x), x) cannot be solved by the factorable group method