14.16.5 problem 19

Internal problem ID [2675]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.9, The method of Laplace transform. Excercises page 232
Problem number : 19
Date solved : Monday, January 27, 2025 at 06:06:15 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+7 y&=\cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 1.036 (sec). Leaf size: 42

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+7*y(t)=cos(t),y(0) = 0, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = \frac {52 \,{\mathrm e}^{-\frac {3 t}{2}} \sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )}{285}-\frac {2 \,{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {\sqrt {19}\, t}{2}\right )}{15}+\frac {\sin \left (t \right )}{15}+\frac {2 \cos \left (t \right )}{15} \]

✓ Solution by Mathematica

Time used: 1.382 (sec). Leaf size: 63

DSolve[{D[y[t],{t,2}]+3*D[y[t],t]+7*y[t]==Cos[t],{y[0]==0,Derivative[1][y][0] ==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{285} \left (19 \sin (t)+52 \sqrt {19} e^{-3 t/2} \sin \left (\frac {\sqrt {19} t}{2}\right )+38 \cos (t)-38 e^{-3 t/2} \cos \left (\frac {\sqrt {19} t}{2}\right )\right ) \]