76.15.28 problem 29

Internal problem ID [17669]
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 : 29
Date solved : Tuesday, January 28, 2025 at 10:50:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&={\mathrm e}^{t} \left (t^{2}+1\right ) \sin \left (2 t \right )+3 \,{\mathrm e}^{-t} \cos \left (t \right )+4 \,{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 73

dsolve(diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=exp(t)*(1+t^2)*sin(2*t)+3*exp(-t)*cos(t)+4*exp(t),y(t), singsol=all)
 

\[ y = -\frac {5 \left (\left (\left (t^{2}-\frac {73}{65} t +\frac {821}{676}\right ) \cos \left (t \right )^{2}-\frac {\sin \left (t \right ) \left (t^{2}+\frac {40}{13} t -\frac {1233}{676}\right ) \cos \left (t \right )}{5}-\frac {t^{2}}{2}+\frac {73 t}{130}-\frac {82619}{20280}\right ) {\mathrm e}^{3 t}-\frac {26 c_{2} {\mathrm e}^{t}}{5}+\frac {39 \,{\mathrm e}^{t} \cos \left (t \right )}{5}-\frac {39 \,{\mathrm e}^{t} \sin \left (t \right )}{5}+\frac {26 c_{1}}{5}\right ) {\mathrm e}^{-2 t}}{26} \]

✓ Solution by Mathematica

Time used: 0.617 (sec). Leaf size: 109

DSolve[D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==Exp[t]*(1+t^2)*Sin[2*t]+3*Exp[-t]*Cos[t]+4*Exp[t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{52} e^t t^2 \sin (2 t)-\frac {e^t \left (3380 t^2-3796 t+4105\right ) \cos (2 t)}{35152}+\frac {2 e^t}{3}+\frac {10}{169} e^t t \sin (2 t)+\frac {3}{2} e^{-t} \sin (t)-\frac {1233 e^t \sin (2 t)}{35152}-\frac {3}{2} e^{-t} \cos (t)+c_1 e^{-2 t}+c_2 e^{-t} \]