57.9.20 problem 3

Internal problem ID [14428]
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 2.3.1 Nonhomogeneous Equations: Undetermined Coefficients. Exercises page 110
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:37:04 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }-b 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 )&=0 \\ \end{align*}

✓ Maple. Time used: 0.186 (sec). Leaf size: 135

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

\[ x = \frac {\left (-\sqrt {b^{2}-4}\, b^{2}-b^{3}-6 \sqrt {b^{2}-4}+4 b \right ) {\mathrm e}^{-\frac {\left (-b +\sqrt {b^{2}-4}\right ) t}{2}}+\left (\sqrt {b^{2}-4}\, b^{2}-b^{3}+6 \sqrt {b^{2}-4}+4 b \right ) {\mathrm e}^{\frac {\left (b +\sqrt {b^{2}-4}\right ) t}{2}}+2 \left (b^{3}-4 b \right ) \cos \left (2 t \right )+3 \left (-b^{2}+4\right ) \sin \left (2 t \right )}{4 b^{4}-7 b^{2}-36} \]

✓ Mathematica. Time used: 0.283 (sec). Leaf size: 122

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

\begin{align*} x(t)&\to \frac {\frac {e^{\frac {1}{2} \left (b-\sqrt {b^2-4}\right ) t} \left (b^2 \left (e^{\sqrt {b^2-4} t}-1\right )-\sqrt {b^2-4} b \left (e^{\sqrt {b^2-4} t}+1\right )+6 e^{\sqrt {b^2-4} t}-6\right )}{\sqrt {b^2-4}}+2 b \cos (2 t)-3 \sin (2 t)}{4 b^2+9} \end{align*}

✓ Sympy. Time used: 0.226 (sec). Leaf size: 236

from sympy import * 
t = symbols("t") 
b = symbols("b") 
x = Function("x") 
ode = Eq(-b*Derivative(x(t), t) + x(t) - sin(2*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)
 

\[ x{\left (t \right )} = \frac {2 b \cos {\left (2 t \right )}}{4 b^{2} + 9} + \left (- \frac {b^{2}}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}} - \frac {b \sqrt {b^{2} - 4}}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}} - \frac {6}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}}\right ) e^{\frac {t \left (b - \sqrt {b^{2} - 4}\right )}{2}} + \left (\frac {b^{2}}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}} - \frac {b \sqrt {b^{2} - 4}}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}} + \frac {6}{4 b^{2} \sqrt {b^{2} - 4} + 9 \sqrt {b^{2} - 4}}\right ) e^{\frac {t \left (b + \sqrt {b^{2} - 4}\right )}{2}} - \frac {3 \sin {\left (2 t \right )}}{4 b^{2} + 9} \]