72.21.8 problem 8

Internal problem ID [14899]
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 6.6. page 624
Problem number : 8
Date solved : Thursday, March 13, 2025 at 05:19:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+16 y&=t \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 8.661 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+16*y(t) = t; 
ic:=y(0) = 1, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \cos \left (4 t \right )+\frac {15 \sin \left (4 t \right )}{64}+\frac {t}{16} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 24

ode=D[y[t],{t,2}]+16*y[t]==t; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{64} (4 t+15 \sin (4 t))+\cos (4 t) \]

✓ Sympy. Time used: 0.086 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t + 16*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t}{16} + \frac {15 \sin {\left (4 t \right )}}{64} + \cos {\left (4 t \right )} \]