Internal problem ID [9416]
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.
CAS Maple gives this as type [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }-a^{2} \left (\left (y^{\prime }\right )^{5}+2 \left (y^{\prime }\right )^{3}+y^{\prime }\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (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 \relax (x ) = \int \RootOf \left (3 \left (\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} \right )+x +c_{2}\right )d x +c_{3} \\ y \relax (x ) = \int \RootOf \left (-3 \left (\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} \right )+x +c_{2}\right )d x +c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 10.187 (sec). Leaf size: 145
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 \\ \end{align*}