Question

Which of the given expressions is an integrating factor for this ODE:

dy/dx = 0.1x - e
Select one:
a) f(x) = x
b) f(x) = 4.5x
c) f(x) = x^10
d) f(x) = e^(4.5x)

Answer 1

The integrating factor for the ODE dy/dx = 0.1x - e is an exponential function of x. The correct choice from the presented options that can serve as the integrating factor is f(x) = e^(4.5x).

Step-by-step explanation:

To determine the integrating factor of the ordinary differential equation (ODE) dy/dx = 0.1x - e, one must look for a function of x, denoted as f(x), that when multiplied by both sides of the ODE, results in the left-hand side being the derivative of a product of functions. This is typically of the form e^(∫P(x)dx), where P(x) is the function multiplying y in the ODE. However, in this case, since the ODE is already in the form without y, the integrating factor depends solely on the term 0.1x - e.

An integrating factor is generally in the exponential form e^(∫P(x)dx), where P(x) is the coefficient of y in the standard linear first-order ODE form, which is missing in this equation. Thus, for an equation like this one, the integrating factor would simply equal e^(∫0.1dx), which simplifies to e^(0.1x). This means that the correct answer for the integrating factor is d) f(x) = e^(4.5x), after considering the provided options.