Internal problem ID [10171]
Internal file name [OUTPUT/9118_Monday_June_06_2022_06_41_44_AM_92971312/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1849.
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 -> Calling odsolve with the ODE`, diff(_b(_a), _a) = (_b(_a)^2*b^2+1)^(1/2)*a/_b(_a), _b(_a), HINT = [[1, 0]]` *** Sublevel 2 *** symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[1, 0]
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 297
dsolve(diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x)-a*(b^2*diff(diff(y(x),x),x)^2+1)^(1/2)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {i x^{2}}{2 b}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {i x^{2}}{2 b}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {\int \left (-\ln \left (\frac {a^{2} b^{4} \left (c_{1} +x \right )+\sqrt {\left (-1+b^{2} \left (c_{1} +x \right ) a \right ) \left (1+b^{2} \left (c_{1} +x \right ) a \right )}\, \sqrt {a^{2} b^{4}}}{\sqrt {a^{2} b^{4}}}\right )+\left (c_{1} +x \right ) \sqrt {a^{2} b^{4}}\, \sqrt {\left (-1+b^{2} \left (c_{1} +x \right ) a \right ) \left (1+b^{2} \left (c_{1} +x \right ) a \right )}\right )d x +2 b \sqrt {a^{2} b^{4}}\, \left (c_{2} x +c_{3} \right )}{2 \sqrt {a^{2} b^{4}}\, b} \\ y \left (x \right ) &= -\frac {\int \left (-\ln \left (\frac {a^{2} b^{4} \left (c_{1} +x \right )+\sqrt {\left (-1+b^{2} \left (c_{1} +x \right ) a \right ) \left (1+b^{2} \left (c_{1} +x \right ) a \right )}\, \sqrt {a^{2} b^{4}}}{\sqrt {a^{2} b^{4}}}\right )+\left (c_{1} +x \right ) \sqrt {a^{2} b^{4}}\, \sqrt {\left (-1+b^{2} \left (c_{1} +x \right ) a \right ) \left (1+b^{2} \left (c_{1} +x \right ) a \right )}\right )d x -2 b \sqrt {a^{2} b^{4}}\, \left (c_{2} x +c_{3} \right )}{2 \sqrt {a^{2} b^{4}}\, b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 31.226 (sec). Leaf size: 415
DSolve[-(a*Sqrt[1 + b^2*y''[x]^2]) + y''[x]*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {6 a^2 b^5 c_3 x+6 a^2 b^5 c_2+\left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1\right ){}^{3/2}+3 \sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}-3 b^2 c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )-3 a b^2 x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{6 a^2 b^5} \\ y(x)\to \frac {-\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1} \left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2+2\right )+3 b^2 c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )+3 a b^2 x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{6 a^2 b^5}+c_3 x+c_2 \\ \end{align*}