72.19.4 problem 30

Internal problem ID [14962]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 6. Laplace transform. Section 6.3 page 600
Problem number : 30
Date solved : Tuesday, January 28, 2025 at 07:25:39 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+13 y&=13 \operatorname {Heaviside}\left (t -4\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 14.105 (sec). Leaf size: 51

dsolve([diff(y(t),t$2)+6*diff(y(t),t)+13*y(t)=13*Heaviside(t-4),y(0) = 3, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = \left (-\frac {1}{2}-\frac {3 i}{4}\right ) \operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{\left (-3-2 i\right ) \left (t -4\right )}+\left (-\frac {1}{2}+\frac {3 i}{4}\right ) \operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{\left (-3+2 i\right ) \left (t -4\right )}+\operatorname {Heaviside}\left (t -4\right )+{\mathrm e}^{-3 t} \left (3 \cos \left (2 t \right )+5 \sin \left (2 t \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 82

DSolve[{D[y[t],{t,2}]-4*D[y[t],t]+5*y[t]==UnitStep[t-4],{y[0]==3,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{2 t} (3 \cos (t)-5 \sin (t)) & t\leq 4 \\ -\frac {1}{5} e^{2 t-8} \cos (4-t)+3 e^{2 t} \cos (t)-\frac {2}{5} e^{2 t-8} \sin (4-t)-5 e^{2 t} \sin (t)+\frac {1}{5} & \text {True} \\ \end {array} \\ \end {array} \]