40.6.11 problem 11

Internal problem ID [8643]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.2, page 216
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 05:40:08 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+\frac {9 y}{4}&=9 t^{3}+64 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&={\frac {63}{2}} \\ \end{align*}

✓ Maple. Time used: 0.109 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+9/4*y(t) = 9*t^3+64; 
ic:=[y(0) = 1, D(y)(0) = 63/2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 4 t^{3}-16 t^{2}+32 t +{\mathrm e}^{-\frac {3 t}{2}} \left (1+t \right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 28

ode=D[y[t],{t,2}]+3*D[y[t],t]+225/100*y[t]==9*t^3+64; 
ic={y[0]==1,Derivative[1][y][0] ==315/10}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 4 t \left (t^2-4 t+8\right )+e^{-3 t/2} (t+1) \end{align*}

✓ Sympy. Time used: 0.132 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-9*t**3 + 9*y(t)/4 + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 64,0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 63/2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 4 t^{3} - 16 t^{2} + 32 t + \left (t + 1\right ) e^{- \frac {3 t}{2}} \]