Internal problem ID [10159]
Internal file name [OUTPUT/9106_Monday_June_06_2022_06_39_36_AM_20048340/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1837.
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, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
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) = a^2*_b(_a)*(_b(_a)^4+2*_b(_a)^2+1), _b(_a), HINT = [[1, 0]]` *** Su symmetry methods on request `, `2nd order, trying reduction of order with given symmetries:`[1, 0]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 105
dsolve(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,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \int \operatorname {RootOf}\left (3 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {3 \textit {\_f}^{6} a^{2}+9 a^{2} \textit {\_f}^{4}+9 \textit {\_f}^{2} a^{2}+3 a^{2}+9 c_{1}}}d \textit {\_f} \right )+x +c_{2} \right )d x +c_{3} \\ y \left (x \right ) &= \int \operatorname {RootOf}\left (-3 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {3 \textit {\_f}^{6} a^{2}+9 a^{2} \textit {\_f}^{4}+9 \textit {\_f}^{2} a^{2}+3 a^{2}+9 c_{1}}}d \textit {\_f} \right )+x +c_{2} \right )d x +c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 22.017 (sec). Leaf size: 442
DSolve[-(a^2*(y'[x] + 2*y'[x]^3 + y'[x]^5)) + Derivative[3][y][x] == 0,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*}