62.3.1 problem 6.3 (a)

Internal problem ID [15435]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 6. Laplace transforms. Problems page 172
Problem number : 6.3 (a)
Date solved : Thursday, October 02, 2025 at 10:14:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+9 y&=18 t \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y \left (\frac {\pi }{2}\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 14

ode:=diff(diff(y(t),t),t)+9*y(t) = 18*t; 
ic:=[y(0) = 0, y(1/2*Pi) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \pi \sin \left (3 t \right )+2 t \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 15

ode=D[y[t],{t,2}]+9*y[t]==18*t; 
ic={y[0]==0,y[Pi/2]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 t+\pi \sin (3 t) \end{align*}

✓ Sympy. Time used: 0.038 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-18*t + 9*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, y(pi/2): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 2 t + \pi \sin {\left (3 t \right )} \]