Internal problem ID [9951]
Internal file name [OUTPUT/8898_Monday_June_06_2022_05_47_48_AM_84734185/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1629 (6.39).
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, _with_potential_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right )=0} \]
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 -> Calling odsolve with the ODE`, diff(diff(diff(y(x), x), x), x) = -f(x)*(diff(diff(y(x), x), x)), y(x)` *** Sublevel 2 *** Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -f(_a)*_b(_a), _b(_a)` *** Sublevel 3 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- high order exact linear fully integrable successful <- differential order: 2; linearization to 3rd order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 38
dsolve(diff(diff(y(x),x),x)+(3*y(x)+f(x))*diff(y(x),x)+y(x)^3+y(x)^2*f(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \left (\int {\mathrm e}^{-\left (\int f \left (x \right )d x \right )}d x \right )+c_{2}}{c_{1} \left (\int \left (\int {\mathrm e}^{-\left (\int f \left (x \right )d x \right )}d x \right )d x \right )+c_{2} x +1} \]
✓ Solution by Mathematica
Time used: 60.065 (sec). Leaf size: 75
DSolve[f[x]*y[x]^2 + y[x]^3 + (f[x] + 3*y[x])*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\int _1^x\exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) c_1dK[2]+c_2}{\int _1^x\int _1^{K[5]}\exp \left (-\int _1^{K[4]}f(K[3])dK[3]\right ) c_1dK[4]dK[5]+c_2 x+1} \]