76.15.29 problem 30

Internal problem ID [17599]
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 : 30
Date solved : Tuesday, April 01, 2025 at 12:00:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=3 t \,{\mathrm e}^{-t} \cos \left (2 t \right )-2 t \,{\mathrm e}^{-2 t} \cos \left (t \right ) \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 64

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

\[ y = \frac {\left (75 t \cos \left (2 t \right ) {\mathrm e}^{t}+\left (-160 t -112\right ) \cos \left (t \right )+\left (80 t +16\right ) \sin \left (t \right )\right ) {\mathrm e}^{-2 t}}{400}+\frac {3 \,{\mathrm e}^{-t} \left (\left (t^{2}+\frac {8 c_2}{3}+6\right ) \sin \left (2 t \right )+\frac {8 c_1 \cos \left (2 t \right )}{3}\right )}{8} \]

✓ Mathematica. Time used: 0.852 (sec). Leaf size: 70

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

\[ y(t)\to \frac {e^{-2 t} \left (25 e^t \left (24 t^2-3+64 c_1\right ) \sin (2 t)+64 (5 t+1) \sin (t)-64 (10 t+7) \cos (t)+100 e^t (3 t+16 c_2) \cos (2 t)\right )}{1600} \]

✓ Sympy. Time used: 0.674 (sec). Leaf size: 58

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

\[ y{\left (t \right )} = \left (\left (C_{1} + \frac {3 t}{16}\right ) \cos {\left (2 t \right )} + \left (C_{2} + \frac {3 t^{2}}{8}\right ) \sin {\left (2 t \right )} + \frac {\left (5 t \sin {\left (t \right )} - 10 t \cos {\left (t \right )} + \sin {\left (t \right )} - 7 \cos {\left (t \right )}\right ) e^{- t}}{25}\right ) e^{- t} \]