76.15.4 problem 4

Internal problem ID [17566]
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 : 4
Date solved : Thursday, March 13, 2025 at 10:13:21 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }&=3+4 \sin \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t) = 3+4*sin(2*t); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.284 (sec). Leaf size: 38

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

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

✓ Sympy. Time used: 0.209 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*sin(2*t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 3,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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