76.22.9 problem 22 (b.1)

Internal problem ID [17711]
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 : Thursday, March 13, 2025 at 10:47:52 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*}

✓ Maple. Time used: 10.282 (sec). Leaf size: 15

ode:=7/5*diff(diff(y(t),t),t)+y(t) = Heaviside(t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 0.422 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - Heaviside(t) + 7*Derivative(y(t), (t, 2))/5,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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