70.15.5 problem 5

Internal problem ID [18933]
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, October 02, 2025 at 03:33:07 PM
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: 0.078 (sec). Leaf size: 88

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

\begin{align*} y(t)&\to \cos (3 t) \int _1^t-\frac {1}{3} \left (e^{3 K[1]} K[1]^2+6\right ) \sin (3 K[1])dK[1]+\sin (3 t) \int _1^t\frac {1}{3} \cos (3 K[2]) \left (e^{3 K[2]} K[2]^2+6\right )dK[2]+c_1 \cos (3 t)+c_2 \sin (3 t) \end{align*}

✓ Sympy. Time used: 0.089 (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)
 

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