Internal problem ID [12190]
Internal file name [OUTPUT/10843_Thursday_September_21_2023_05_47_56_AM_22119359/index.tex]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER.
Problems page 172
Problem number: Problem 40(a).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Unable to solve or complete the solution.
\[ \boxed {m x^{\prime \prime }-f \left (x\right )=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)-f(_a)/m = 0, _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 62
dsolve(m*diff(x(t),t$2)=f(x(t)),x(t), singsol=all)
\begin{align*} m \left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {m \left (c_{1} m +2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )\right )}}d \textit {\_b} \right )-t -c_{2} &= 0 \\ -m \left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {m \left (c_{1} m +2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )\right )}}d \textit {\_b} \right )-t -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.049 (sec). Leaf size: 44
DSolve[m*x''[t]==f[x[t]],x[t],t,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{x(t)}\frac {1}{\sqrt {c_1+2 \int _1^{K[2]}\frac {f(K[1])}{m}dK[1]}}dK[2]{}^2=(t+c_2){}^2,x(t)\right ] \]