76.15.29 problem 30

Internal problem ID [17670]
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, January 28, 2025 at 10:52:04 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*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 54

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

\[ y = \frac {3 \left (\left (t^{2}+\frac {8 c_{2}}{3}+6\right ) {\mathrm e}^{t} \sin \left (2 t \right )+\frac {\left (t +\frac {16 c_{1}}{3}\right ) {\mathrm e}^{t} \cos \left (2 t \right )}{2}+\left (-\frac {16 t}{15}-\frac {56}{75}\right ) \cos \left (t \right )+\frac {8 \left (t +\frac {1}{5}\right ) \sin \left (t \right )}{15}\right ) {\mathrm e}^{-2 t}}{8} \]

✓ Solution by Mathematica

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

DSolve[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],y[t],t,IncludeSingularSolutions -> True]
 

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