72.21.2 problem 2

Internal problem ID [14972]
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 6.6. page 624
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 07:25:55 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+5 y&=\operatorname {Heaviside}\left (t -2\right ) \sin \left (4 t -8\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 13.284 (sec). Leaf size: 100

dsolve([diff(y(t),t$2)+diff(y(t),t)+5*y(t)=Heaviside(t-2)*sin(4*(t-2)),y(0) = -2, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {4 \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-\frac {t}{2}+1}}{137}+\frac {92 \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \sqrt {19}\, \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-\frac {t}{2}+1}}{2603}-2 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {19}\, t}{2}\right )-\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )}{19}-\frac {4 \left (\cos \left (4 t -8\right )+\frac {11 \sin \left (4 t -8\right )}{4}\right ) \operatorname {Heaviside}\left (t -2\right )}{137} \]

✓ Solution by Mathematica

Time used: 10.086 (sec). Leaf size: 877

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {e^{-t/2} \left (\left (467205706-107188712 \sqrt {19}\right ) \cos \left (\frac {\sqrt {19} t}{2}\right )+2 \left (-53594356+12294887 \sqrt {19}\right ) \sin \left (\frac {\sqrt {19} t}{2}\right )\right )}{-233602853+53594356 \sqrt {19}} & t\leq 2 \\ \frac {e^{-\frac {i \left (\left (83+16 \sqrt {19}\right ) t+16 \left ((8-i)+\sqrt {19}\right )\right )}{2 \left (-8+\sqrt {19}\right )}} \left (\left (137 \left ((-1394672-17233937 i)+(319819+3953224 i) \sqrt {19}\right ) e^{\frac {83 i t+16 i \sqrt {19}+8}{-8+\sqrt {19}}}-137 i \left ((-17233937-1394672 i)+(3953224+319819 i) \sqrt {19}\right ) e^{\frac {8 \left (2 i \sqrt {19} t+(1+16 i)\right )}{-8+\sqrt {19}}}+(32+88 i) \left ((-105794040-17233937 i)+(24269955+3953224 i) \sqrt {19}\right ) e^{\frac {\left ((8+83 i)-(1-16 i) \sqrt {19}\right ) t+(2+32 i) \sqrt {19}+90 i}{2 \left (-8+\sqrt {19}\right )}}-1096 \left (-233602853+53594356 \sqrt {19}\right ) e^{\frac {\left ((8+83 i)-(1-16 i) \sqrt {19}\right ) t+16 \left ((1+8 i)+i \sqrt {19}\right )}{2 \left (-8+\sqrt {19}\right )}}-(88+32 i) \left ((-17233937-105794040 i)+(3953224+24269955 i) \sqrt {19}\right ) e^{\frac {\left ((8+83 i)-(1-16 i) \sqrt {19}\right ) t+2 \left (83 i+\sqrt {19}\right )}{2 \left (-8+\sqrt {19}\right )}}+\left ((2059892888+7500312135 i)-(472570051+1720667520 i) \sqrt {19}\right ) e^{\frac {8 i \left (8+\sqrt {19}\right ) t+16 i \sqrt {19}+8}{-8+\sqrt {19}}}+(11+4 i) \left ((-53594356-662357701 i)+(12294887+151953452 i) \sqrt {19}\right ) e^{\frac {i \left (19+8 \sqrt {19}\right ) t+(8+128 i)}{-8+\sqrt {19}}}\right ) \cos \left (\frac {\sqrt {19} t}{2}\right )+\left (-137 i \left ((-1394672-17233937 i)+(319819+3953224 i) \sqrt {19}\right ) e^{\frac {83 i t+16 i \sqrt {19}+8}{-8+\sqrt {19}}}+137 \left ((-17233937-1394672 i)+(3953224+319819 i) \sqrt {19}\right ) e^{\frac {8 \left (2 i \sqrt {19} t+(1+16 i)\right )}{-8+\sqrt {19}}}-(88-32 i) \left ((-105794040-17233937 i)+(24269955+3953224 i) \sqrt {19}\right ) e^{\frac {\left ((8+83 i)-(1-16 i) \sqrt {19}\right ) t+(2+32 i) \sqrt {19}+90 i}{2 \left (-8+\sqrt {19}\right )}}+1096 \left (-53594356+12294887 \sqrt {19}\right ) e^{\frac {\left ((8+83 i)-(1-16 i) \sqrt {19}\right ) t+16 \left ((1+8 i)+i \sqrt {19}\right )}{2 \left (-8+\sqrt {19}\right )}}-(32-88 i) \left ((-17233937-105794040 i)+(3953224+24269955 i) \sqrt {19}\right ) e^{\frac {\left ((8+83 i)-(1-16 i) \sqrt {19}\right ) t+2 \left (83 i+\sqrt {19}\right )}{2 \left (-8+\sqrt {19}\right )}}+(11-4 i) \left ((-662357701-53594356 i)+(151953452+12294887 i) \sqrt {19}\right ) e^{\frac {8 i \left (8+\sqrt {19}\right ) t+16 i \sqrt {19}+8}{-8+\sqrt {19}}}+(4-11 i) \left ((-53594356-662357701 i)+(12294887+151953452 i) \sqrt {19}\right ) e^{\frac {i \left (19+8 \sqrt {19}\right ) t+(8+128 i)}{-8+\sqrt {19}}}\right ) \sin \left (\frac {\sqrt {19} t}{2}\right )\right )}{548 \left (-233602853+53594356 \sqrt {19}\right )} & \text {True} \\ \end {array} \\ \end {array} \]