87.22.28 problem 28

Internal problem ID [23773]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 4. The Laplace transform. Exercise at page 199
Problem number : 28
Date solved : Thursday, October 02, 2025 at 09:45:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }-3 y&=3 t^{3}-9 t^{2}-5 t +1 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=8 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)-3*y(t) = 3*t^3-9*t^2-5*t+1; 
ic:=[y(0) = 0, D(y)(0) = 8]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{t}+1-t^{3}+t^{2}-2 \,{\mathrm e}^{-3 t}+t \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 26

ode=D[y[t],{t,2}]+2*D[y[t],t]-3*y[t]==3*t^3-9*t^2-5*t+1; 
ic={y[0]==0,Derivative[1][y][0] ==8}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.164 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*t**3 + 9*t**2 + 5*t - 3*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 1,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 8} 
dsolve(ode,func=y(t),ics=ics)
 

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