76.15.5 problem 5

Internal problem ID [17567]
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 : Thursday, March 13, 2025 at 10:13:25 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

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

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

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

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

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

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

✓ Sympy. Time used: 0.156 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2*exp(3*t) + 9*y(t) + Derivative(y(t), (t, 2)) - 6,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} \sin {\left (3 t \right )} + C_{2} \cos {\left (3 t \right )} + \frac {t^{2} e^{3 t}}{18} - \frac {t e^{3 t}}{27} + \frac {e^{3 t}}{162} + \frac {2}{3} \]