Internal problem ID [9796]
Internal file name [OUTPUT/8739_Monday_June_06_2022_05_22_18_AM_63950698/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _fully, _exact, _linear]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime } \sin \left (x \right )-2 y^{\prime } \cos \left (x \right )+y \sin \left (x \right )=\ln \left (x \right )} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }-\left (\frac {d}{d x}y^{\prime }\right ) \sin \left (x \right )-2 y^{\prime } \cos \left (x \right )+y \sin \left (x \right )=\ln \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable <- high order exact linear fully integrable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 41
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)
\[ y \left (x \right ) = \frac {\left (4 c_{3} +\int \left (8 c_{1} x +4 c_{2} -3 x^{2}+2 x^{2} \ln \left (x \right )\right ) {\mathrm e}^{\cos \left (x \right )}d x \right ) {\mathrm e}^{-\cos \left (x \right )}}{4} \]
✓ Solution by Mathematica
Time used: 2.289 (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]
\[ y(x)\to e^{-\cos (x)} \left (\int _1^x\frac {1}{4} e^{\cos (K[1])} \left (2 \log (K[1]) K[1]^2-3 K[1]^2+4 c_1 K[1]+4 c_2\right )dK[1]+c_3\right ) \]