28.3.17 problem 6.52

Internal problem ID [4530]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.52
Date solved : Monday, January 27, 2025 at 09:22:48 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y&=120 \,{\mathrm e}^{3 t} \operatorname {Heaviside}\left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=15\\ y^{\prime }\left (0\right )&=-6\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 4.109 (sec). Leaf size: 73

dsolve([diff(y(t),t$4)-5*diff(y(t),t$2)+4*y(t)=120*exp(3*t)*Heaviside(t-1),y(0) = 15, D(y)(0) = -6, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

\[ y = -10 \,{\mathrm e}^{-2 t} \left (-{\mathrm e}^{3 t +2} \operatorname {Heaviside}\left (-1+t \right )+{\mathrm e}^{4 t +1} \operatorname {Heaviside}\left (-1+t \right )+\left (-\frac {{\mathrm e}^{5}}{5}-\frac {3 \,{\mathrm e}^{5 t}}{10}+\frac {{\mathrm e}^{t +4}}{2}\right ) \operatorname {Heaviside}\left (-1+t \right )-\frac {7 \,{\mathrm e}^{t}}{5}-\frac {3 \,{\mathrm e}^{3 t}}{5}+\frac {{\mathrm e}^{4 t}}{5}+\frac {3}{10}\right ) \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 95

DSolve[{D[y[t],{t,4}]-5*D[y[t],{t,2}]+4*y[t]==120*Exp[3*t]*UnitStep[t-1],{y[0]==15,Derivative[1][y][0] == -6,Derivative[2][y][0] == 0,Derivative[3][y][0] == 0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-2 t} \left (-\left (\left (3 e^t+2 e\right ) \left (e-e^t\right )^4 \theta (1-t)\right )+14 e^t+6 e^{3 t}-2 e^{4 t}+3 e^{5 t}-5 e^{t+4}+10 e^{3 t+2}-10 e^{4 t+1}+2 e^5-3\right ) \]