76.15.26 problem 27

Internal problem ID [17667]
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 : 27
Date solved : Tuesday, January 28, 2025 at 10:50:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 48

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

\[ y = \frac {3}{4}+\frac {\left (2 t^{3}+3 c_{1} t +3 c_{2} \right ) {\mathrm e}^{2 t}}{3}+\frac {\left (2 t +1\right ) \cos \left (2 t \right )}{16}+\frac {t^{2}}{2}+t -\frac {\sin \left (2 t \right )}{16} \]

✓ Solution by Mathematica

Time used: 0.439 (sec). Leaf size: 67

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

\[ y(t)\to \frac {2}{3} e^{2 t} t^3+\frac {t^2}{2}+t+\frac {1}{16} (2 t+1) \cos (2 t)+c_2 e^{2 t} t+c_1 e^{2 t}-\frac {1}{8} \sin (t) \cos (t)+\frac {3}{4} \]