73.3.4 problem 7 (iv)

Internal problem ID [19814]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 5. Linear equations. Exercises at page 85
Problem number : 7 (iv)
Date solved : Thursday, October 02, 2025 at 04:44:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 46

ode:=diff(diff(x(t),t),t)-diff(x(t),t)+x(t) = sin(2*t); 
ic:=[x(0) = 0, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {40 \,{\mathrm e}^{\frac {t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{39}-\frac {2 \,{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )}{13}-\frac {3 \sin \left (2 t \right )}{13}+\frac {2 \cos \left (2 t \right )}{13} \]

✓ Mathematica. Time used: 1.197 (sec). Leaf size: 67

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

\begin{align*} x(t)&\to \frac {1}{39} \left (-9 \sin (2 t)+40 \sqrt {3} e^{t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )+6 \cos (2 t)-6 e^{t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.161 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) - sin(2*t) - Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {40 \sqrt {3} \sin {\left (\frac {\sqrt {3} t}{2} \right )}}{39} - \frac {2 \cos {\left (\frac {\sqrt {3} t}{2} \right )}}{13}\right ) e^{\frac {t}{2}} - \frac {3 \sin {\left (2 t \right )}}{13} + \frac {2 \cos {\left (2 t \right )}}{13} \]