60.3.412 problem 1418

Internal problem ID [11422]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1418
Date solved : Monday, January 27, 2025 at 11:20:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {x \sin \left (x \right ) y^{\prime }}{\cos \left (x \right ) x -\sin \left (x \right )}+\frac {\sin \left (x \right ) y}{\cos \left (x \right ) x -\sin \left (x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.546 (sec). Leaf size: 47

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)
 

\[ y = \sin \left (x \right ) \left (c_{1} +c_{2} \left (\int {\mathrm e}^{-\int \frac {2 \cos \left (x \right ) \cot \left (x \right ) x -3 \cos \left (x \right )+\sec \left (x \right )}{-\sin \left (x \right )+\cos \left (x \right ) x}d x} \cos \left (x \right )d x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.144 (sec). Leaf size: 15

DSolve[D[y[x],{x,2}] == (Sin[x]*y[x])/(x*Cos[x] - Sin[x]) - (x*Sin[x]*D[y[x],x])/(x*Cos[x] - Sin[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 x+c_2 \sin (x) \]