76.20.13 problem 13

Internal problem ID [17761]
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 5.6 (Differential equations with Discontinuous Forcing Functions). Problems at page 342
Problem number : 13
Date solved : Tuesday, January 28, 2025 at 11:02:26 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+4 y&=1-\operatorname {Heaviside}\left (t -\pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 12.201 (sec). Leaf size: 27

dsolve([diff(y(t),t$4)+5*diff(y(t),t$2)+4*y(t)=1-Heaviside(t-Pi),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\left (\cos \left (t \right )+1\right )^{2} \operatorname {Heaviside}\left (t -\pi \right )}{6}+\frac {\left (\cos \left (t \right )-1\right )^{2}}{6} \]

✓ Solution by Mathematica

Time used: 2.293 (sec). Leaf size: 14575

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

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