32.4.13 problem 14

Internal problem ID [7786]
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 : Tuesday, September 30, 2025 at 05:05:26 PM
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: 0.047 (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,op(ic)],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 90

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]
 

\begin{align*} x(t)&\to -e^t \left (-\int _1^t-e^{-K[1]} \sin (K[1])dK[1]+e^t \left (\int _1^0e^{-2 K[2]} \sin (K[2])dK[2]-\int _1^te^{-2 K[2]} \sin (K[2])dK[2]\right )+\int _1^0-e^{-K[1]} \sin (K[1])dK[1]\right ) \end{align*}

✓ Sympy. Time used: 0.130 (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)
 

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