(dy/dx) + 8y =5

To solve the differential equation (dy/dx) + 8y = 5, we can use the method of integrating factors.

Step 1: Identify the form of the equation

The given differential equation is in the form of a first-order linear ordinary differential equation. It has the form (dy/dx) + P(x)y = Q(x), where P(x) = 8 and Q(x) = 5.

Step 2: Find the integrating factor

The key step in solving a first-order linear ordinary differential equation is to find the integrating factor, which is denoted by the symbol μ(x). In this case, the integrating factor μ(x) can be found by multiplying both sides of the equation by the exponential of the integral of P(x):

μ(x) = e^(∫P(x)dx)

In our case, we have P(x) = 8. Therefore, the integrating factor μ(x) can be found as:

μ(x) = e^(∫8dx)

Evaluating the integral, we have:

μ(x) = e^(8x)

Step 3: Multiply both sides by the integrating factor

Now, we multiply both sides of the equation (dy/dx) + 8y = 5 by the integrating factor μ(x):

e^(8x)(dy/dx) + 8e^(8x)y = 5e^(8x)

Step 4: Simplify and integrate

The left-hand side of the equation can be simplified using the product rule of differentiation:

(d/dx)[e^(8x)y] = 5e^(8x)

Now, we integrate both sides with respect to x:

∫(d/dx)[e^(8x)y] dx = ∫5e^(8x) dx

Integrating, we obtain:

e^(8x)y = (5/8)e^(8x) + C

where C is the constant of integration.

Step 5: Solve for y

To solve for y, we divide both sides of the equation by e^(8x):

y = (5/8)e^(8x) / e^(8x) + C / e^(8x)

Simplifying, we get:

y = (5/8) + Ce^(-8x)

where C is the constant of integration.

Therefore, the general solution to the given differential equation is y = (5/8) + Ce^(-8x).