Internal problem ID [9049]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1470.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _fully, _exact, _linear]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }-y^{\prime \prime } \sin \relax (x )-2 y^{\prime } \cos \relax (x )+y \sin \relax (x )-\ln \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 36
dsolve(diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)*sin(x)-2*diff(y(x),x)*cos(x)+y(x)*sin(x)-ln(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \left (c_{3}+\int \left (2 x c_{1}+c_{2}-\frac {3 x^{2}}{4}+\frac {x^{2} \ln \relax (x )}{2}\right ) {\mathrm e}^{\cos \relax (x )}d x \right ) {\mathrm e}^{-\cos \relax (x )} \]
✓ Solution by Mathematica
Time used: 1.615 (sec). Leaf size: 57
DSolve[-Log[x] + Sin[x]*y[x] - 2*Cos[x]*y'[x] - Sin[x]*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\cos (x)} \left (\int _1^x\frac {1}{4} e^{\cos (K[1])} \left (2 \log (K[1]) K[1]^2-3 K[1]^2+4 c_1 K[1]+4 c_2\right )dK[1]+c_3\right ) \\ \end{align*}