Internal problem ID [10169]
Internal file name [OUTPUT/9116_Monday_June_06_2022_06_41_23_AM_33934950/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1847.
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], [_3rd_order, _reducible, _mu_poly_yn]]
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 -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = 3*(diff(_b(_a), _a))^2*_b(_a)/(_b(_a)^2+1), _b(_a), HINT = [[1, 0], [ symmetry methods on request `, `2nd order, trying reduction of order with given symmetries:`[1, 0], [_a, 0]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 67
dsolve((diff(y(x),x)^2+1)*diff(diff(diff(y(x),x),x),x)-3*diff(y(x),x)*diff(diff(y(x),x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= -\sqrt {-c_{2}^{2}-2 c_{2} x -x^{2}+c_{1}}+c_{3} \\ y \left (x \right ) &= \sqrt {-c_{2}^{2}-2 c_{2} x -x^{2}+c_{1}}+c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.928 (sec). Leaf size: 142
DSolve[-3*y'[x]*y''[x]^2 + (1 + y'[x]^2)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_3-\frac {i \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}{c_1} \\ y(x)\to \frac {i \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}{c_1}+c_3 \\ y(x)\to \text {Indeterminate} \\ y(x)\to c_3-i \sqrt {(x+c_2){}^2} \\ y(x)\to i \sqrt {(x+c_2){}^2}+c_3 \\ \end{align*}