Internal problem ID [10176]
Internal file name [OUTPUT/9123_Monday_June_06_2022_06_42_39_AM_27363095/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1854.
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.
\[ \boxed {y^{\prime \prime }-f \left (y\right )=0} \]
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) = 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: 51
dsolve(diff(y(x),x$2)-f(y(x))=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )+c_{1}}}d \textit {\_b} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )+c_{1}}}d \textit {\_b} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.047 (sec). Leaf size: 40
DSolve[-f[y[x]]+ y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\sqrt {c_1+2 \int _1^{K[2]}f(K[1])dK[1]}}dK[2]{}^2=(x+c_2){}^2,y(x)\right ] \]