68.11.28 problem 40

Internal problem ID [17565]
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 : 40
Date solved : Thursday, October 02, 2025 at 02:25:29 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 37

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

\[ y = -\frac {{\mathrm e}^{3 t} \left (\left (t^{4}-4 t^{3}+12 t^{2}-2 c_1 -24 t \right ) {\mathrm e}^{t}-2 c_2 \right )}{2} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 42

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

\begin{align*} y(t)&\to e^{3 t} \left (e^t \left (-\frac {t^4}{2}+2 t^3-6 t^2+12 t-12+c_2\right )+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.203 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*t**3*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)
 

\[ y{\left (t \right )} = \left (C_{1} + \left (C_{2} - \frac {t^{4}}{2} + 2 t^{3} - 6 t^{2} + 12 t\right ) e^{t}\right ) e^{3 t} \]