72.19.6 problem 32

Internal problem ID [14893]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 6. Laplace transform. Section 6.3 page 600
Problem number : 32
Date solved : Monday, March 31, 2025 at 01:01:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y&=\operatorname {Heaviside}\left (t -4\right ) \cos \left (5 t -20\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=-2 \end{align*}

✓ Maple. Time used: 0.402 (sec). Leaf size: 42

ode:=diff(diff(y(t),t),t)+3*y(t) = Heaviside(t-4)*cos(5*t-20); 
ic:=y(0) = 0, D(y)(0) = -2; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -4\right ) \cos \left (5 t -20\right )}{22}+\frac {\operatorname {Heaviside}\left (t -4\right ) \cos \left (\sqrt {3}\, \left (t -4\right )\right )}{22}-\frac {2 \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )}{3} \]

✓ Mathematica. Time used: 1.481 (sec). Leaf size: 202

ode=D[y[t],{t,2}]+3*y[t]==UnitStep[t-4]*Cos[5*(t-4)]; 
ic={y[0]==0,Derivative[1][y][0] ==-2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {i e^{-i \sqrt {3} t} \left (-1+e^{2 i \sqrt {3} t}\right )}{\sqrt {3}} & t\leq 4 \\ \frac {1}{132} e^{-i \left (5+\sqrt {3}\right ) (t+4)} \left (\cos \left (\sqrt {3} t\right )+i \sin \left (\sqrt {3} t\right )\right ) \left (-3 e^{4 i \sqrt {3}} \left (e^{40 i}+e^{10 i t}\right )+3 e^{5 i (t+4)} \left (1+e^{8 i \sqrt {3}}\right ) \cos \left (\sqrt {3} t\right )-e^{5 i (t+4)} \left (-3 i+88 \sqrt {3} e^{4 i \sqrt {3}}+3 i e^{8 i \sqrt {3}}\right ) \sin \left (\sqrt {3} t\right )\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 72.963 (sec). Leaf size: 1788

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*y(t) - cos(5*t - 20)*Heaviside(t - 4) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): -2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \text {Solution too large to show} \]