Internal problem ID [5885]
Internal file name [OUTPUT/5133_Sunday_June_05_2022_03_25_46_PM_97727503/index.tex]
Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS.
K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.6 Summary and Problems. Page
218
Problem number: Problem 3.18.
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) = (1+_b(_a)^2)^(1/2)/(_b(_a)*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: 0.141 (sec). Leaf size: 175
dsolve(a*diff(y(x),x$2)*diff(y(x),x$3)=sqrt(1+ diff(y(x),x$2)^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {1}{2} i x^{2}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {1}{2} i x^{2}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {\left (2 a^{2}+\left (x +c_{1} \right )^{2}\right ) \sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}-3 \left (a \left (x +c_{1} \right ) \ln \left (c_{1} +x +\sqrt {\left (c_{1} +a +x \right ) \left (c_{1} -a +x \right )}\right )-2 c_{2} x -2 c_{3} \right ) a}{6 a} \\ y \left (x \right ) &= \frac {\left (-2 a^{2}-\left (x +c_{1} \right )^{2}\right ) \sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}+3 a \left (a \left (x +c_{1} \right ) \ln \left (c_{1} +x +\sqrt {\left (c_{1} +a +x \right ) \left (c_{1} -a +x \right )}\right )+2 c_{2} x +2 c_{3} \right )}{6 a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 11.484 (sec). Leaf size: 209
DSolve[a*y''[x]*y'''[x]==Sqrt[1+ y''[x]^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2} \left (a^2 \left (2+c_1{}^2\right )+2 a c_1 x+x^2\right )}{6 a}-\frac {1}{2} a (x+a c_1) \log \left (\sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2}+a c_1+x\right )+c_3 x+c_2 \\ y(x)\to -\frac {\sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2} \left (a^2 \left (2+c_1{}^2\right )+2 a c_1 x+x^2\right )}{6 a}+\frac {1}{2} a (x+a c_1) \log \left (\sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2}+a c_1+x\right )+c_3 x+c_2 \\ \end{align*}