Internal problem ID [15409]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 15.5 Linear equations with variable coefficients.
The Lagrange method. Exercises page 148
Problem number: 640.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (\tan \left (x \right )-2 \cot \left (x \right )\right ) y^{\prime }+2 \cot \left (x \right )^{2} y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (x \right ) \end {align*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 13
dsolve([diff(y(x),x$2)+(tan(x)-2*cot(x))*diff(y(x),x)+2*cot(x)^2*y(x)=0,sin(x)],singsol=all)
\[ y \left (x \right ) = \sin \left (x \right ) \left (\sin \left (x \right ) c_{2} +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 2.22 (sec). Leaf size: 27
DSolve[y''[x]+(Tan[x]-2*Cot[x])*y'[x]+2*Cot[x]^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \sqrt {-\sin ^2(x)}-c_2 \sin ^2(x) \]