Internal problem ID [10165]
Internal file name [OUTPUT/9112_Monday_June_06_2022_06_40_52_AM_88810108/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1843.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
Unable to parse ODE.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying 3rd order ODE linearizable_by_differentiation differential order: 3; trying a linearization to 4th order trying differential order: 3; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, (diff(diff(_b(_a), _a), _a))*_b(_a)^2-_b(_a)*(-_a*(diff(_b(_a), _a))^2-_a^3+(diff(_b(_a), _a))*_b( symmetry methods on request `, `2nd order, trying reduction of order with given symmetries:`[_a, 2*_b]
✓ Solution by Maple
Time used: 0.093 (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 \left (x \right ) &= 0 \\ -2 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\textit {\_a}^{4}+4 \textit {\_a}^{2} c_{2} -4 c_{2}^{2}+4 c_{1}}}d \textit {\_a} \right )-x -c_{3} &= 0 \\ 2 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\textit {\_a}^{4}+4 \textit {\_a}^{2} c_{2} -4 c_{2}^{2}+4 c_{1}}}d \textit {\_a} \right )-x -c_{3} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.269 (sec). Leaf size: 409
DSolve[y[x]^3*y'[x] - y'[x]*y''[x] + 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*}