60.8.7 problem 1843
Internal
problem
ID
[11842]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
7,
non-linear
third
and
higher
order
Problem
number
:
1843
Date
solved
:
Tuesday, January 28, 2025 at 06:23:53 PM
CAS
classification
:
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]
\begin{align*} y y^{\prime \prime \prime }-y^{\prime } y^{\prime \prime }+y^{3} y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.108 (sec). Leaf size: 81
dsolve(y(x)*diff(diff(diff(y(x),x),x),x)-diff(y(x),x)*diff(diff(y(x),x),x)+y(x)^3*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
-2 \left (\int _{}^{y}\frac {1}{\sqrt {-\textit {\_a}^{4}+4 c_{2} \textit {\_a}^{2}-4 c_{2}^{2}+4 c_{1}}}d \textit {\_a} \right )-x -c_3 &= 0 \\
2 \left (\int _{}^{y}\frac {1}{\sqrt {-\textit {\_a}^{4}+4 c_{2} \textit {\_a}^{2}-4 c_{2}^{2}+4 c_{1}}}d \textit {\_a} \right )-x -c_3 &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.520 (sec). Leaf size: 409
DSolve[y[x]^3*D[y[x],x] - D[y[x],x]*D[y[x],{x,2}] + y[x]*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {2 i \sqrt {1+\frac {\text {$\#$1}^2}{2 \left (\sqrt {c_2{}^2-c_1}-c_2\right )}} \sqrt {1-\frac {\text {$\#$1}^2}{2 \left (c_2+\sqrt {c_2{}^2-c_1}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \text {$\#$1}}{\sqrt {2}}\right ),\frac {c_2-\sqrt {c_2{}^2-c_1}}{c_2+\sqrt {c_2{}^2-c_1}}\right )}{\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \sqrt {-\frac {\text {$\#$1}^4}{2}+2 \text {$\#$1}^2 c_2-2 c_1}}\&\right ][x+c_3] \\
y(x)\to \text {InverseFunction}\left [\frac {2 i \sqrt {1+\frac {\text {$\#$1}^2}{2 \left (\sqrt {c_2{}^2-c_1}-c_2\right )}} \sqrt {1-\frac {\text {$\#$1}^2}{2 \left (c_2+\sqrt {c_2{}^2-c_1}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \text {$\#$1}}{\sqrt {2}}\right ),\frac {c_2-\sqrt {c_2{}^2-c_1}}{c_2+\sqrt {c_2{}^2-c_1}}\right )}{\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \sqrt {-\frac {\text {$\#$1}^4}{2}+2 \text {$\#$1}^2 c_2-2 c_1}}\&\right ][x+c_3] \\
\end{align*}