76.22.2 problem 15

Internal problem ID [17783]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.8 (Convolution Integrals and Their Applications). Problems at page 359
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 11:02:59 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+25 y&=\sin \left (\alpha t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 12.026 (sec). Leaf size: 65

dsolve([diff(y(t),t$2)+6*diff(y(t),t)+25*y(t)=sin(alpha*t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\left (24 \cos \left (4 t \right )+\sin \left (4 t \right ) \left (\alpha ^{2}-7\right )\right ) \alpha \,{\mathrm e}^{-3 t}-4 \alpha ^{2} \sin \left (\alpha t \right )-24 \alpha \cos \left (\alpha t \right )+100 \sin \left (\alpha t \right )}{4 \alpha ^{4}-56 \alpha ^{2}+2500} \]

✓ Solution by Mathematica

Time used: 1.535 (sec). Leaf size: 126

DSolve[{D[y[t],{t,2}]+D[y[t],t]+25*y[t]==Sin[a*t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {e^{-t/2} \left (2 \sqrt {11} a^3 \sin \left (\frac {3 \sqrt {11} t}{2}\right )-33 a^2 e^{t/2} \sin (a t)-49 \sqrt {11} a \sin \left (\frac {3 \sqrt {11} t}{2}\right )+825 e^{t/2} \sin (a t)+33 a \cos \left (\frac {3 \sqrt {11} t}{2}\right )-33 a e^{t/2} \cos (a t)\right )}{33 \left (a^4-49 a^2+625\right )} \]