Internal problem ID [9853]
Internal file name [OUTPUT/8798_Monday_June_06_2022_05_30_29_AM_32380306/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1529.
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.
\[ \boxed {\left (\sin \left (x \right )+x \right ) y^{\prime \prime \prime }+3 \left (\cos \left (x \right )+1\right ) y^{\prime \prime }-3 y^{\prime } \sin \left (x \right )-y \cos \left (x \right )=-\sin \left (x \right )} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\sin \left (x \right )+x \right ) \left (\frac {d}{d x}y^{\prime \prime }\right )+3 \left (\cos \left (x \right )+1\right ) \left (\frac {d}{d x}y^{\prime }\right )-3 y^{\prime } \sin \left (x \right )-y \cos \left (x \right )=-\sin \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
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: 25
dsolve((sin(x)+x)*diff(diff(diff(y(x),x),x),x)+3*(cos(x)+1)*diff(diff(y(x),x),x)-3*diff(y(x),x)*sin(x)-y(x)*cos(x)+sin(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{3} +c_{1} x^{2}+x c_{2} -\cos \left (x \right )}{\sin \left (x \right )+x} \]
✓ Solution by Mathematica
Time used: 0.113 (sec). Leaf size: 28
DSolve[Sin[x] - Cos[x]*y[x] - 3*Sin[x]*y'[x] + 3*(1 + Cos[x])*y''[x] + (x + Sin[x])*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {-\cos (x)+x (c_3 x+c_2)+c_1}{x+\sin (x)} \]