65.10.4 problem 17.4

Internal problem ID [13787]
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 : 17.4
Date solved : Tuesday, January 28, 2025 at 06:03:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-t^{2}+t \right ) x^{\prime \prime }+\left (-t^{2}+2\right ) x^{\prime }+\left (2-t \right ) x&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} x&={\mathrm e}^{-t} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 17

dsolve([(t-t^2)*diff(x(t),t$2)+(2-t^2)*diff(x(t),t)+(2-t)*x(t)=0,exp(-t)],singsol=all)
 

\[ x \left (t \right ) = \frac {c_{2} {\mathrm e}^{-t} t +c_{1}}{t} \]

✓ Solution by Mathematica

Time used: 0.224 (sec). Leaf size: 96

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

\[ x(t)\to \exp \left (\int _1^t\frac {K[1]-2}{2 (K[1]-1)}dK[1]-\frac {1}{2} \int _1^t\left (\frac {2}{K[2]}+1+\frac {1}{1-K[2]}\right )dK[2]\right ) \left (c_2 \int _1^t\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]\right )dK[3]+c_1\right ) \]