Internal problem ID [9051]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1472.
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 ) \left (x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.02 (sec). Leaf size: 33
dsolve(diff(diff(diff(y(x),x),x),x)+f(x)*(x^2*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+2*y(x))=0,y(x), singsol=all)
\[ y \relax (x ) = \left (\int \left (c_{1}+c_{2} \left (\int {\mathrm e}^{-\left (\int \left (f \relax (x ) x^{2}+\frac {3}{x}\right )d x \right )}d x \right )\right )d x +c_{3}\right ) x \]
✓ Solution by Mathematica
Time used: 0.068 (sec). Leaf size: 85
DSolve[f[x]*(2*y[x] - 2*x*y'[x] + x^2*y''[x]) + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \left (c_3 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}f(K[1]) K[1]^2dK[1]\right )}{K[2]^2}dK[2]-x \int _1^x\frac {\exp \left (-\int _1^{K[3]}f(K[1]) K[1]^2dK[1]\right )}{K[3]^3}dK[3]\right )+c_2 x+c_1\right ) \\ \end{align*}