11.4.5 problem 5

Internal problem ID [1499]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing functions. page 268
Problem number : 5
Date solved : Monday, January 27, 2025 at 04:58:05 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}&=t -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (t -\frac {\pi }{2}\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: 0.396 (sec). Leaf size: 63

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

\[ y = -\frac {16}{25}-\frac {12 \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )+\frac {4 \sin \left (t \right )}{3}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}}{25}+\frac {2 \left (8-10 t +5 \pi \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )}{25}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}} \left (4 \cos \left (t \right )-3 \sin \left (t \right )\right )}{25}+\frac {4 t}{5} \]

✓ Solution by Mathematica

Time used: 0.036 (sec). Leaf size: 96

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {4}{25} e^{-t/2} \left (e^{t/2} (5 t-4)+4 \cos (t)-3 \sin (t)\right ) & 2 t\leq \pi \\ -\frac {2}{25} e^{-t/2} \left (\left (-8+6 e^{\pi /4}\right ) \cos (t)+\left (6+8 e^{\pi /4}\right ) \sin (t)-5 e^{t/2} \pi \right ) & \text {True} \\ \end {array} \\ \end {array} \]