76.15.25 problem 26

Internal problem ID [17666]
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 : Tuesday, January 28, 2025 at 10:50:04 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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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