Internal problem ID [9362]
Internal file name [OUTPUT/8298_Monday_June_06_2022_02_40_04_AM_58658032/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1029.
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_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }-\left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients <- unable to find a useful change of variables trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: trying Riccati_symmetries <- Riccati particular solution successful <- reduction of order to Riccati successful <- reduction of order to Riccati successful`
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 22
dsolve(diff(diff(y(x),x),x)-(f(x)^2+diff(f(x),x))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\int {\mathrm e}^{-2 \left (\int f \left (x \right )d x \right )}d x +c_{1} \right ) {\mathrm e}^{\int f \left (x \right )d x} c_{2} \]
✓ Solution by Mathematica
Time used: 0.053 (sec). Leaf size: 58
DSolve[-(y[x]*(f[x]^2 + Derivative[1][f][x])) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \exp \left (\int _1^xf(K[1])dK[1]\right )+c_2 \exp \left (\int _1^xf(K[2])dK[2]\right ) \int _1^x\exp \left (\int _1^{K[4]}-2 f(K[3])dK[3]\right )dK[4] \]