72.19.5 problem 31

Internal problem ID [14884]
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 : 31
Date solved : Thursday, March 13, 2025 at 05:18:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\cos \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 7.898 (sec). Leaf size: 18

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

\[ y = -2 \cos \left (2 t \right )+\frac {\sin \left (2 t \right ) t}{4} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 94

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

\[ y(t)\to -\sin (2 t) \int _1^0\frac {1}{2} \cos ^2(2 K[2])dK[2]+\sin (2 t) \int _1^t\frac {1}{2} \cos ^2(2 K[2])dK[2]+\cos (2 t) \left (\int _1^t-\frac {1}{4} \sin (4 K[1])dK[1]-\int _1^0-\frac {1}{4} \sin (4 K[1])dK[1]-2\right ) \]

✓ Sympy. Time used: 0.104 (sec). Leaf size: 17

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

\[ y{\left (t \right )} = \frac {t \sin {\left (2 t \right )}}{4} - 2 \cos {\left (2 t \right )} \]