(dy/dx) + 8y =5

To solve the given differential equation, you can use the method of integrating factors.

Step 1: Write the differential equation in the standard form:
(dy/dx) + 8y = 5

Step 2: Identify the coefficient of y, which is 8.

Step 3: Multiply the entire equation by the integrating factor, which is defined as e^(∫(coefficient of y)dx). In this case, the integrating factor is e^(8x).

Multiplying both sides of the equation by e^(8x), we get:
e^(8x)(dy/dx) + 8e^(8x)y = 5e^(8x)

Step 4: Recognize that the left side of the equation can be written as the derivative of the product of the integrating factor and the dependent variable y. In this case, the left side can be rewritten as:

(d/dx)(e^(8x)y)

Applying the derivative to the left side, we have:
d/dx(e^(8x)y) = 5e^(8x)

Now, we have a separable differential equation.

Step 5: Integrate both sides of the equation with respect to x:
∫d/dx(e^(8x)y) dx = ∫5e^(8x) dx

Integration of the left side yields:
e^(8x)y = ∫5e^(8x) dx
e^(8x)y = (5/8)e^(8x) + C

Here, C is the constant of integration.

Step 6: Solve for y by dividing both sides of the equation by e^(8x):
y = [(5/8)e^(8x) + C] / e^(8x)
y = (5/8) + Ce^(-8x)

Finally, the solution to the given differential equation is y = (5/8) + Ce^(-8x), where C is a constant.