63.13.4 problem 5

Internal problem ID [13175]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.3 Reduction of order. Exercises page 125
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 05:12:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} x&=t \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 12

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

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 17

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

\[ x(t)\to e t \left (c_2 e^t+c_1\right ) \]