Internal problem ID [9107]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1528.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_y]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime } \sin \relax (x )+\left (2 \cos \relax (x )+1\right ) y^{\prime \prime }-y^{\prime } \sin \relax (x )-\cos \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 68
dsolve(diff(diff(diff(y(x),x),x),x)*sin(x)+(2*cos(x)+1)*diff(diff(y(x),x),x)-diff(y(x),x)*sin(x)-cos(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\ln \left (\sin \relax (x )\right ) \sin \relax (x )-\ln \left (-\frac {\cos \relax (x )-1}{\sin \relax (x )}\right ) \sin \relax (x )-\cos \relax (x ) x +x \right ) c_{1}}{\cos \relax (x )-1}+c_{2}+\frac {\sin \relax (x ) c_{3}}{\cos \relax (x )-1}+\frac {-\cos \relax (x ) x +\sin \relax (x )-x}{\sin \relax (x )} \]
✓ Solution by Mathematica
Time used: 0.414 (sec). Leaf size: 56
DSolve[-Cos[x] - Sin[x]*y'[x] + (1 + 2*Cos[x])*y''[x] + Sin[x]*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \cot \left (\frac {x}{2}\right ) \left (2 \text {ArcSin}(\cos (x))-\sqrt {2} (c_2 \log (2 (\cos (x)+1))+2 c_1)\right )-\frac {c_2 x}{\sqrt {2}}+c_3 \\ \end{align*}