76.15.25 problem 26

Internal problem ID [17587]
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 : 26
Date solved : Thursday, March 13, 2025 at 10:16:12 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 43

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

\[ y = -\frac {2 \,{\mathrm e}^{-t} \left (-\frac {9}{2}+\left (t^{3}-\frac {3}{2} t -\frac {3}{2} c_{1} -\frac {3}{2}\right ) \cos \left (t \right )+\frac {3 \left (-t^{2}-c_{2} -t +1\right ) \sin \left (t \right )}{2}\right )}{3} \]

✓ Mathematica. Time used: 0.209 (sec). Leaf size: 50

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

\[ y(t)\to \frac {1}{6} e^{-t} \left (\left (-4 t^3+6 t+6+6 c_2\right ) \cos (t)+\left (6 t^2+6 t-3+6 c_1\right ) \sin (t)+18\right ) \]

✓ Sympy. Time used: 0.492 (sec). Leaf size: 31

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

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