68.11.20 problem 32

Internal problem ID [17557]
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 : 32
Date solved : Thursday, October 02, 2025 at 02:25:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+8 y&=-256 t^{3} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=diff(diff(y(t),t),t)-6*diff(y(t),t)+8*y(t) = -256*t^3; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {{\mathrm e}^{4 t} c_1}{2}-45-84 t -72 t^{2}-32 t^{3}+{\mathrm e}^{2 t} c_2 \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 36

ode=D[y[t],{t,2}]-6*D[y[t],t]+8*y[t]==-256*t^3; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -32 t^3-72 t^2-84 t+c_1 e^{2 t}+c_2 e^{4 t}-45 \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(256*t**3 + 8*y(t) - 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{4 t} - 32 t^{3} - 72 t^{2} - 84 t - 45 \]