76.22.9 problem 22 (b.1)

Internal problem ID [17790]
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 5.8 (Convolution Integrals and Their Applications). Problems at page 359
Problem number : 22 (b.1)
Date solved : Tuesday, January 28, 2025 at 11:03:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \frac {7 y^{\prime \prime }}{5}+y&=\operatorname {Heaviside}\left (t \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: 10.282 (sec). Leaf size: 15

dsolve([diff(y(t),t$2)+2*(1/5)*diff(y(t),t$2)+y(t)=Heaviside(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = 1-\cos \left (\frac {\sqrt {35}\, t}{7}\right ) \]

✓ Solution by Mathematica

Time used: 0.046 (sec). Leaf size: 58

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

\[ y(t)\to -\frac {1}{12} e^{-t/5} \theta (t) \left (-12 e^{t/5}+\sqrt {6} \sin \left (\frac {2 \sqrt {6} t}{5}\right )+12 \cos \left (\frac {2 \sqrt {6} t}{5}\right )\right ) \]