Internal problem ID [9050]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1471.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }+f \relax (x ) y^{\prime \prime }+y^{\prime }+f \relax (x ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 36
dsolve(diff(diff(diff(y(x),x),x),x)+f(x)*diff(diff(y(x),x),x)+diff(y(x),x)+f(x)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{i x} \left (\int {\mathrm e}^{-2 i x} \left (\int c_{3} {\mathrm e}^{\int \left (i-f \relax (x )\right )d x}d x +c_{2}\right )d x +c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.065 (sec). Leaf size: 78
DSolve[f[x]*y[x] + y'[x] + f[x]*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{i x} \left (c_3 \int _1^xe^{-2 i K[3]} \int _1^{K[3]}\exp \left (\int _1^{K[2]}(i-f(K[1]))dK[1]\right )dK[2]dK[3]+c_1\right )+\frac {1}{2} c_2 (\sin (x)+i \cos (x)) \\ \end{align*}