76.15.5 problem 5

Internal problem ID [17646]
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 : 5
Date solved : Tuesday, January 28, 2025 at 10:47:20 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=t^{2} {\mathrm e}^{3 t}+6 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 29

dsolve(diff(y(t),t$2)+9*y(t)=t^2*exp(3*t)+6,y(t), singsol=all)
 

\[ y = \sin \left (3 t \right ) c_{2} +\cos \left (3 t \right ) c_{1} +\frac {2}{3}+\frac {\left (t -\frac {1}{3}\right )^{2} {\mathrm e}^{3 t}}{18} \]

✓ Solution by Mathematica

Time used: 0.221 (sec). Leaf size: 50

DSolve[D[y[t],{t,2}]+9*y[t]==t^2*Exp[3*t]+6,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{162} \left (9 e^{3 t} t^2-6 e^{3 t} t+e^{3 t}+162 c_1 \cos (3 t)+162 c_2 \sin (3 t)+108\right ) \]