65.10.6 problem 17.6

Internal problem ID [13789]
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
Date solved : Tuesday, January 28, 2025 at 06:03:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \tan \left (t \right ) x^{\prime \prime }-3 x^{\prime }+\left (\tan \left (t \right )+3 \cot \left (t \right )\right ) x&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} x&=\sin \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.367 (sec). Leaf size: 13

dsolve([tan(t)*diff(x(t),t$2)-3*diff(x(t),t)+(tan(t)+3*cot(t))*x(t)=0,sin(t)],singsol=all)
 

\[ x \left (t \right ) = \sin \left (t \right ) \left (c_{1} +c_{2} \cos \left (t \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.187 (sec). Leaf size: 24

DSolve[Tan[t]*D[x[t],{t,2}]-3*D[x[t],t]+(Tan[t]+3*Cot[t])*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \sqrt {-\sin ^2(t)} (c_2 \cos (t)+c_1) \]