60.8.1 problem 1837
Internal
problem
ID
[11762]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
7,
non-linear
third
and
higher
order
Problem
number
:
1837
Date
solved
:
Sunday, March 30, 2025 at 09:14:09 PM
CAS
classification
:
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
\begin{align*} y^{\prime \prime \prime }-a^{2} \left ({y^{\prime }}^{5}+2 {y^{\prime }}^{3}+y^{\prime }\right )&=0 \end{align*}
✓ Maple. Time used: 0.030 (sec). Leaf size: 105
ode:=diff(diff(diff(y(x),x),x),x)-a^2*(diff(y(x),x)^5+2*diff(y(x),x)^3+diff(y(x),x)) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \int \operatorname {RootOf}\left (3 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {3 \textit {\_f}^{6} a^{2}+9 \textit {\_f}^{4} a^{2}+9 \textit {\_f}^{2} a^{2}+3 a^{2}+9 c_1}}d \textit {\_f} +x +c_2 \right )d x +c_3 \\
y &= \int \operatorname {RootOf}\left (-3 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {3 \textit {\_f}^{6} a^{2}+9 \textit {\_f}^{4} a^{2}+9 \textit {\_f}^{2} a^{2}+3 a^{2}+9 c_1}}d \textit {\_f} +x +c_2 \right )d x +c_3 \\
\end{align*}
✓ Mathematica. Time used: 21.385 (sec). Leaf size: 442
ode=-(a^2*(D[y[x],x] + 2*D[y[x],x]^3 + D[y[x],x]^5)) + Derivative[3][y][x] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \int _1^x\text {InverseFunction}\left [-3 \int \frac {1}{\sqrt {3 \left (a^2\right )^2 \text {$\#$1}^6+9 \left (a^2\right )^2 \text {$\#$1}^4+9 \left (a^2\right )^2 \text {$\#$1}^2+9 c_1}}d\text {$\#$1}\&\right ][c_2-K[1]]dK[1]+c_3 \\
y(x)\to \int _1^x\text {InverseFunction}\left [3 \int \frac {1}{\sqrt {3 \left (a^2\right )^2 \text {$\#$1}^6+9 \left (a^2\right )^2 \text {$\#$1}^4+9 \left (a^2\right )^2 \text {$\#$1}^2+9 c_1}}d\text {$\#$1}\&\right ][c_2-K[2]]dK[2]+c_3 \\
y(x)\to \text {Indeterminate} \\
y(x)\to \int _1^x\text {InverseFunction}\left [-3 \int \frac {1}{\sqrt {3 \left (a^2\right )^2 \text {$\#$1}^6+9 \left (a^2\right )^2 \text {$\#$1}^4+9 \left (a^2\right )^2 \text {$\#$1}^2+9 (-1) c_1}}d\text {$\#$1}\&\right ][c_2-K[1]]dK[1]+c_3 \\
y(x)\to \int _1^x\text {InverseFunction}\left [3 \int \frac {1}{\sqrt {3 \left (a^2\right )^2 \text {$\#$1}^6+9 \left (a^2\right )^2 \text {$\#$1}^4+9 \left (a^2\right )^2 \text {$\#$1}^2+9 (-1) c_1}}d\text {$\#$1}\&\right ][c_2-K[2]]dK[2]+c_3 \\
y(x)\to \int _1^x\text {InverseFunction}\left [-3 \int \frac {1}{\sqrt {3 \left (a^2\right )^2 \text {$\#$1}^6+9 \left (a^2\right )^2 \text {$\#$1}^4+9 \left (a^2\right )^2 \text {$\#$1}^2+9 c_1}}d\text {$\#$1}\&\right ][c_2-K[1]]dK[1]+c_3 \\
y(x)\to \int _1^x\text {InverseFunction}\left [3 \int \frac {1}{\sqrt {3 \left (a^2\right )^2 \text {$\#$1}^6+9 \left (a^2\right )^2 \text {$\#$1}^4+9 \left (a^2\right )^2 \text {$\#$1}^2+9 c_1}}d\text {$\#$1}\&\right ][c_2-K[2]]dK[2]+c_3 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*(Derivative(y(x), x)**5 + 2*Derivative(y(x), x)**3 + Derivative(y(x), x)) + Derivative(y(x), (x, 3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out