63.9.20 problem 3

Internal problem ID [13147]
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 : Tuesday, January 28, 2025 at 05:07:53 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*}

✓ Solution by Maple

Time used: 0.228 (sec). Leaf size: 135

dsolve([diff(x(t),t$2)-b*diff(x(t),t)+x(t)=sin(2*t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.339 (sec). Leaf size: 241

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

\[ x(t)\to -e^{\frac {1}{2} \left (b-\sqrt {b^2-4}\right ) t} \int _1^0-\frac {e^{\frac {1}{2} \left (\sqrt {b^2-4}-b\right ) K[1]} \sin (2 K[1])}{\sqrt {b^2-4}}dK[1]+e^{\frac {1}{2} \left (b-\sqrt {b^2-4}\right ) t} \int _1^t-\frac {e^{\frac {1}{2} \left (\sqrt {b^2-4}-b\right ) K[1]} \sin (2 K[1])}{\sqrt {b^2-4}}dK[1]+e^{\frac {1}{2} \left (\sqrt {b^2-4}+b\right ) t} \left (\int _1^t\frac {e^{-\frac {1}{2} \left (b+\sqrt {b^2-4}\right ) K[2]} \sin (2 K[2])}{\sqrt {b^2-4}}dK[2]-\int _1^0\frac {e^{-\frac {1}{2} \left (b+\sqrt {b^2-4}\right ) K[2]} \sin (2 K[2])}{\sqrt {b^2-4}}dK[2]\right ) \]