76.22.6 problem 19

Internal problem ID [17787]
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 : 19
Date solved : Tuesday, January 28, 2025 at 11:03:04 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\cos \left (\alpha t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 10.892 (sec). Leaf size: 76

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=cos(alpha*t),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\left (-\alpha ^{4}-3 \alpha ^{2}-2\right ) {\mathrm e}^{-2 t}+\left (2 \alpha ^{4}+9 \alpha ^{2}+4\right ) {\mathrm e}^{-t}-\cos \left (\alpha t \right ) \alpha ^{2}+3 \alpha \sin \left (\alpha t \right )+2 \cos \left (\alpha t \right )}{\left (\alpha ^{2}+4\right ) \left (\alpha ^{2}+1\right )} \]

✓ Solution by Mathematica

Time used: 0.094 (sec). Leaf size: 84

DSolve[{D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==Cos[a*t],{y[0]==1,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {e^{-2 t} \left (2 a^4 e^t-a^4+9 a^2 e^t-\left (a^2-2\right ) e^{2 t} \cos (a t)-3 a^2+3 a e^{2 t} \sin (a t)+4 e^t-2\right )}{a^4+5 a^2+4} \]