48.1.2 problem 2

Internal problem ID [9816]
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 : 2
Date solved : Tuesday, September 30, 2025 at 06:42:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&={\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)-6*diff(y(x),x)+9*y(x) = exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{3 x} \left (c_1 x +c_2 \right )+\frac {{\mathrm e}^{x}}{4} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 50

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

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

✓ Sympy. Time used: 0.115 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \left (\left (C_{1} + C_{2} x\right ) e^{2 x} + \frac {1}{4}\right ) e^{x} \]