77.1.95 problem 122 (page 179)
Internal
problem
ID
[17914]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
122
(page
179)
Date
solved
:
Monday, March 31, 2025 at 04:50:03 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}-{y^{\prime }}^{4}&=0 \end{align*}
✓ Maple. Time used: 0.081 (sec). Leaf size: 105
ode:=y(x)*diff(diff(y(x),x),x)-diff(y(x),x)^2-diff(y(x),x)^4 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
c_1 \ln \left (\frac {c_1 \left (\sqrt {c_1^{2}-y^{2}}+c_1 \right )}{y}\right )+c_1 \ln \left (2\right )-\sqrt {c_1^{2}-y^{2}}-c_2 -x &= 0 \\
-c_1 \ln \left (\frac {c_1 \left (\sqrt {c_1^{2}-y^{2}}+c_1 \right )}{y}\right )-c_1 \ln \left (2\right )+\sqrt {c_1^{2}-y^{2}}-c_2 -x &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.858 (sec). Leaf size: 361
ode=y[x]*D[y[x],{x,2}]-D[y[x],x]^2-D[y[x],x]^4==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-i e^{-(-c_1)} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [i e^{-(-c_1)} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\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)**4 - Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(sqrt(4*y(x)*Derivative(y(x), (x, 2)) + 1)/2 - 1/2) + Derivative(y(x), x) cannot be solved by the factorable group method