76.15.24 problem 25

Internal problem ID [17665]
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 : 25
Date solved : Tuesday, January 28, 2025 at 10:49:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+6 y&={\mathrm e}^{t} \cos \left (2 t \right )+{\mathrm e}^{2 t} \left (3 t +4\right ) \sin \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 50

dsolve(diff(y(t),t$2)-5*diff(y(t),t)+6*y(t)=exp(t)*cos(2*t)+exp(2*t)*(3*t+4)*sin(t),y(t), singsol=all)
 

\[ y = \frac {3 \,{\mathrm e}^{t} \left (\frac {2 c_{2} {\mathrm e}^{2 t}}{3}+\left (\left (t +\frac {1}{3}\right ) \cos \left (t \right )+\left (-t -\frac {10}{3}\right ) \sin \left (t \right )+\frac {2 c_{1}}{3}\right ) {\mathrm e}^{t}-\frac {\cos \left (t \right )^{2}}{15}-\frac {\sin \left (t \right ) \cos \left (t \right )}{5}+\frac {1}{30}\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.580 (sec). Leaf size: 68

DSolve[D[y[t],{t,2}]-5*D[y[t],t]+6*y[t]==Exp[t]*Cos[2*t]+Exp[2*t]*(3*t+4)*Sin[t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{20} e^t \left (-100 e^t \sin (t)-30 e^t t \sin (t)-3 \sin (2 t)+10 e^t (3 t+1) \cos (t)-\cos (2 t)+20 c_1 e^t+20 c_2 e^{2 t}\right ) \]