74.18.27 problem 33

Internal problem ID [16505]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 33
Date solved : Thursday, March 13, 2025 at 08:15:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+7 y^{\prime }+12 y&=3 t^{2} {\mathrm e}^{-4 t} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 31

ode:=diff(diff(y(t),t),t)+7*diff(y(t),t)+12*y(t) = 3*t^2*exp(-4*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (-t^{3}-3 t^{2}+c_{1} -6 t \right ) {\mathrm e}^{-4 t}+{\mathrm e}^{-3 t} c_{2} \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 48

ode=D[y[t],{t,2}]+7*D[y[t],t]+12*y[t]==3*t^2*Exp[-4*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-4 t} \left (e^t \int _1^t3 e^{-K[1]} K[1]^2dK[1]-t^3+c_2 e^t+c_1\right ) \]

✓ Sympy. Time used: 0.336 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*t**2*exp(-4*t) + 12*y(t) + 7*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} + \left (C_{2} - t^{3} - 3 t^{2} - 6 t\right ) e^{- t}\right ) e^{- 3 t} \]