Internal problem ID [10170]
Internal file name [OUTPUT/9117_Monday_June_06_2022_06_41_30_AM_97141835/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1848.
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*_b(_a)+a)*(diff(_b(_a), _a))^2/(_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: 377
dsolve((diff(y(x),x)^2+1)*diff(diff(diff(y(x),x),x),x)-(3*diff(y(x),x)+a)*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 ) &= \int \tan \left (\operatorname {RootOf}\left (c_{2}^{2} a^{4} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} a^{4} x \,{\mathrm e}^{2 \textit {\_Z} a}+a^{4} x^{2} {\mathrm e}^{2 \textit {\_Z} a}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z} a} c_{1} c_{2} a^{3}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z} a} c_{1} a^{3} x +\cos \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}+2 c_{2}^{2} a^{2} {\mathrm e}^{2 \textit {\_Z} a}+4 c_{2} a^{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+2 a^{2} x^{2} {\mathrm e}^{2 \textit {\_Z} a}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z} a} c_{1} c_{2} a -2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z} a} c_{1} a x -\sin \left (\textit {\_Z} \right )^{2} c_{1}^{2}+c_{2}^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+x^{2} {\mathrm e}^{2 \textit {\_Z} a}\right )\right )d x +c_{3} \\ y \left (x \right ) &= \int \tan \left (\operatorname {RootOf}\left (c_{2}^{2} a^{4} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} a^{4} x \,{\mathrm e}^{2 \textit {\_Z} a}+a^{4} x^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z} a} c_{1} c_{2} a^{3}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z} a} c_{1} a^{3} x +\cos \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}+2 c_{2}^{2} a^{2} {\mathrm e}^{2 \textit {\_Z} a}+4 c_{2} a^{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+2 a^{2} x^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z} a} c_{1} c_{2} a +2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z} a} c_{1} a x -\sin \left (\textit {\_Z} \right )^{2} c_{1}^{2}+c_{2}^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+x^{2} {\mathrm e}^{2 \textit {\_Z} a}\right )\right )d x +c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 30.105 (sec). Leaf size: 198
DSolve[(-a - 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 {\left (1-i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \arctan (\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\right ){}^{-\frac {1}{2}-\frac {i a}{2}} \left (1+i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \arctan (\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\right ){}^{\frac {1}{2} i (a+i)} \left (1+a \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \arctan (\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\right )}{\left (a^2+1\right ) c_1} \\ y(x)\to \text {Indeterminate} \\ y(x)\to c_3 \\ \end{align*}