68.11.23 problem 35

Internal problem ID [17560]
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 : 35
Date solved : Thursday, October 02, 2025 at 02:25:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-9 y&=54 t \sin \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 30

ode:=diff(diff(y(t),t),t)-9*y(t) = 54*t*sin(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-3 t} c_2 +{\mathrm e}^{3 t} c_1 -\frac {216 \cos \left (2 t \right )}{169}-\frac {54 t \sin \left (2 t \right )}{13} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 40

ode=D[y[t],{t,2}]-9*y[t]==54*t*Sin[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to c_1 e^{3 t}+c_2 e^{-3 t}-\frac {54}{169} (13 t \sin (2 t)+4 \cos (2 t)) \end{align*}

✓ Sympy. Time used: 0.073 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-54*t*sin(2*t) - 9*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{3 t} - \frac {54 t \sin {\left (2 t \right )}}{13} - \frac {216 \cos {\left (2 t \right )}}{169} \]