38.4.13 problem 14

Internal problem ID [6499]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 25. Second order differential equations. Further problems 25. page 1094
Problem number : 14
Date solved : Wednesday, March 05, 2025 at 12:53:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }-3 x^{\prime }+2 x&=\sin \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 23

ode:=diff(diff(x(t),t),t)-3*diff(x(t),t)+2*x(t) = sin(t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {{\mathrm e}^{2 t}}{5}+\frac {3 \cos \left (t \right )}{10}+\frac {\sin \left (t \right )}{10}-\frac {{\mathrm e}^{t}}{2} \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 27

ode=D[x[t],{t,2}]-3*D[x[t],t]+2*x[t]==Sin[t]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{10} \left (e^t \left (2 e^t-5\right )+\sin (t)+3 \cos (t)\right ) \]

✓ Sympy. Time used: 0.224 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(2*x(t) - sin(t) - 3*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {e^{2 t}}{5} - \frac {e^{t}}{2} + \frac {\sin {\left (t \right )}}{10} + \frac {3 \cos {\left (t \right )}}{10} \]