67.9.26 problem 14.3 (b)

Internal problem ID [16574]
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 (b)
Date solved : Thursday, October 02, 2025 at 01:36:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+8 y&={\mathrm e}^{4 x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)-6*diff(y(x),x)+8*y(x) = exp(4*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{2 x} \left (\left (-\frac {1}{2}+c_1 +x \right ) {\mathrm e}^{2 x}+2 c_2 \right )}{2} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 31

ode=D[y[x],{x,2}]-6*D[y[x],x]+8*y[x]==Exp[4*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^{2 x}+e^{4 x} \left (\frac {x}{2}-\frac {1}{4}+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.137 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*y(x) - exp(4*x) - 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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