76.15.2 problem 2

Internal problem ID [17564]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 2
Date solved : Thursday, March 13, 2025 at 10:13:03 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=3 \sin \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 3*sin(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\sin \left (2 t \right ) \left (17 c_{2} {\mathrm e}^{-t}+3\right )}{17}+\cos \left (2 t \right ) {\mathrm e}^{-t} c_{1} -\frac {12 \cos \left (2 t \right )}{17} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 45

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==3*Sin[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{17} e^{-t} \left (\left (-12 e^t+17 c_2\right ) \cos (2 t)+\left (3 e^t+17 c_1\right ) \sin (2 t)\right ) \]

✓ Sympy. Time used: 0.228 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) - 3*sin(2*t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )}\right ) e^{- t} + \frac {3 \sin {\left (2 t \right )}}{17} - \frac {12 \cos {\left (2 t \right )}}{17} \]