Internal problem ID [11733]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 17, Reduction of order. Exercises page 162
Problem number: 17.6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\tan \left (t \right ) x^{\prime \prime }-3 x^{\prime }+\left (\tan \left (t \right )+3 \cot \left (t \right )\right ) x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= \sin \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 15
dsolve([tan(t)*diff(x(t),t$2)-3*diff(x(t),t)+(tan(t)+3*cot(t))*x(t)=0,sin(t)],x(t), singsol=all)
\[ x \left (t \right ) = c_{1} \sin \left (t \right )+c_{2} \sin \left (t \right ) \cos \left (t \right ) \]
✓ Solution by Mathematica
Time used: 0.374 (sec). Leaf size: 24
DSolve[Tan[t]*x''[t]-3*x'[t]+(Tan[t]+3*Cot[t])*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \sqrt {-\sin ^2(t)} (c_2 \cos (t)+c_1) \]