Internal problem ID [10167]
Internal file name [OUTPUT/9114_Monday_June_06_2022_06_41_08_AM_6502752/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1845.
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, _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 Try integration with the canonical coordinates of the symmetry [0, y] -> Calling odsolve with the ODE`, (4/9)*_b(_a)^3-2*(diff(_b(_a), _a))*_b(_a)+diff(diff(_b(_a), _a), _a) = 0, _b(_a), explicit, HINT symmetry methods on request `, `2nd order, trying reduction of order with given symmetries:`[1, 0], [_a, -_b]
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 85
dsolve(9*y(x)^2*diff(diff(diff(y(x),x),x),x)-45*y(x)*diff(y(x),x)*diff(diff(y(x),x),x)+40*diff(y(x),x)^3=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (-6 \left (\int _{}^{\textit {\_Z}}\frac {1}{4 \textit {\_h}^{2}+\sqrt {c_{1} \left (4 \textit {\_h}^{2}+c_{1} \right )}+c_{1}}d \textit {\_h} \right )+x +c_{2} \right )d x +c_{3}} \\ y \left (x \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (6 \left (\int _{}^{\textit {\_Z}}-\frac {1}{4 \textit {\_h}^{2}-\sqrt {c_{1} \left (4 \textit {\_h}^{2}+c_{1} \right )}+c_{1}}d \textit {\_h} \right )+x +c_{2} \right )d x +c_{3}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.199 (sec). Leaf size: 21
DSolve[40*y'[x]^3 - 45*y[x]*y'[x]*y''[x] + 9*y[x]^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{(x (c_3 x+c_2)+c_1){}^{3/2}} \]