68.11.27 problem 39

Internal problem ID [17564]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 39
Date solved : Thursday, October 02, 2025 at 02:25:28 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }-5 y&=-648 t^{2} {\mathrm e}^{5 t} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 31

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)-5*y(t) = -648*t^2*exp(5*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (-36 t^{3}+18 t^{2}+c_2 -6 t \right ) {\mathrm e}^{5 t}+{\mathrm e}^{-t} c_1 \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 47

ode=D[y[t],{t,2}]-4*D[y[t],t]-5*y[t]==-648*t^2*Exp[5*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} \left (\int _1^t108 e^{6 K[1]} K[1]^2dK[1]+e^{6 t} \left (-36 t^3+c_2\right )+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(648*t**2*exp(5*t) - 5*y(t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} e^{- t} + \left (C_{1} - 36 t^{3} + 18 t^{2} - 6 t\right ) e^{5 t} \]