Internal problem ID [8997]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1418.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {x \sin \relax (x ) y^{\prime }}{\cos \relax (x ) x -\sin \relax (x )}-\frac {\sin \relax (x ) y}{\cos \relax (x ) x -\sin \relax (x )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 61
dsolve(diff(diff(y(x),x),x) = -x*sin(x)/(cos(x)*x-sin(x))*diff(y(x),x)+sin(x)/(cos(x)*x-sin(x))*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \relax (x )+c_{2} \sin \relax (x ) \left (\int {\mathrm e}^{\int \frac {-2 \left (\cos ^{3}\relax (x )\right ) x +3 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-\sin \relax (x )}{\cos \relax (x ) \left (\cos \relax (x ) x -\sin \relax (x )\right ) \sin \relax (x )}d x} \cos \relax (x )d x \right ) \]
✓ Solution by Mathematica
Time used: 0.088 (sec). Leaf size: 15
DSolve[y''[x] == (Sin[x]*y[x])/(x*Cos[x] - Sin[x]) - (x*Sin[x]*y'[x])/(x*Cos[x] - Sin[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x+c_2 \sin (x) \\ \end{align*}