48.1.3 problem 3

Internal problem ID [9817]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 8. Nonhomogeneous Equations: Undetermined Coefficients. Exercises Page 142
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 06:42:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=12 x^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)+2*y(x) = 12*x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -{\mathrm e}^{-2 x} c_1 +{\mathrm e}^{-x} c_2 +6 x^{2}-18 x +21 \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 31

ode=D[y[x],{x,2}]+3*D[y[x],x]+2*y[x]==12*x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 6 x^2-18 x+c_1 e^{-2 x}+c_2 e^{-x}+21 \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-12*x**2 + 2*y(x) + 3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- 2 x} + C_{2} e^{- x} + 6 x^{2} - 18 x + 21 \]