73.20.2 problem 29.6 (b)

Internal problem ID [15540]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 29. Convolution. Additional Exercises. page 523
Problem number : 29.6 (b)
Date solved : Monday, March 31, 2025 at 01:40:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=t \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: 0.088 (sec). Leaf size: 14

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

\[ y = \frac {t}{4}-\frac {\sin \left (2 t \right )}{8} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 17

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

\[ y(t)\to \frac {1}{4} (t-\sin (t) \cos (t)) \]

✓ Sympy. Time used: 0.083 (sec). Leaf size: 12

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

\[ y{\left (t \right )} = \frac {t}{4} - \frac {\sin {\left (2 t \right )}}{8} \]