63.12.6 problem 1(f)

Internal problem ID [13089]
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 2.4.2 Variation of parameters. Exercises page 124
Problem number : 1(f)
Date solved : Wednesday, March 05, 2025 at 09:17:09 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }-2 x^{\prime }+x&=\frac {{\mathrm e}^{t}}{2 t} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 23

ode:=diff(diff(x(t),t),t)-2*diff(x(t),t)+x(t) = 1/2/t*exp(t); 
dsolve(ode,x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 29

ode=D[x[t],{t,2}]-2*D[x[t],t]+x[t]==1/(2*t)*Exp[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{2} e^t (t \log (t)+(-1+2 c_2) t+2 c_1) \]

✓ Sympy. Time used: 0.249 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) - 2*Derivative(x(t), t) + Derivative(x(t), (t, 2)) - exp(t)/(2*t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (C_{1} + t \left (C_{2} + \frac {\log {\left (t \right )}}{2}\right )\right ) e^{t} \]