67.9.25 problem 14.3 (a)

Internal problem ID [16573]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.3 (a)
Date solved : Thursday, October 02, 2025 at 01:36:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+3 y&=9 \,{\mathrm e}^{2 x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{3 x} \end{align*}

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

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+3*y(x) = 9*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} \left (c_2 \,{\mathrm e}^{2 x}+c_1 -9 \,{\mathrm e}^{x}\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 25

ode=D[y[x],{x,2}]-4*D[y[x],x]+3*y[x]==9*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x \left (-9 e^x+c_2 e^{2 x}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*y(x) - 9*exp(2*x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} e^{2 x} - 9 e^{x}\right ) e^{x} \]