67.4.16 problem Problem 3(b)

Internal problem ID [14060]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 3(b)
Date solved : Tuesday, January 28, 2025 at 06:13:21 AM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 9.269 (sec). Leaf size: 38

dsolve([diff(y(t),t)-2*y(t)=4*t*(Heaviside(t)-Heaviside(t-2)),y(0) = 1],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.491 (sec). Leaf size: 75

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

\[ y(t)\to e^{2 t-4} \left (\theta (2-t) \left (e^2 \theta (t) \left (e^2 \int _1^t4 e^{-2 K[1]} K[1]dK[1]+2 e^2-3\right )-2 e^4+5\right )-e^4 \theta (t)+3 e^4-5\right ) \]