(dy/dx) + 8y =5
To solve the given first-order linear differential equation, we will use the method of integrating factor. The equation is in the form of:
(dy/dx) + p(x)y = q(x)
where p(x) = 8 and q(x) = 5.
Step 1: Identify p(x) and q(x)
In this equation, p(x) is the coefficient of y (8y), and q(x) is the constant term (5).
Step 2: Find the integrating factor (IF)
The integrating factor (IF) is defined as the exponential of the integral of p(x) dx. In this case, p(x) is 8, so we find the integral of 8 dx:
∫8 dx = 8x
Therefore, the integrating factor (IF) is e^(8x).
Step 3: Multiply both sides of the equation by the integrating factor (IF)
Multiply both sides of the differential equation by e^(8x):
e^(8x) * (dy/dx) + 8e^(8x) * y = 5e^(8x)
Step 4: Rewrite the left side of the equation using the product rule
The product rule states that (d/dx)(uv) = u * (dv/dx) + v * (du/dx). Applying this rule to the left side of the equation, we have:
(d/dx)(e^(8x) * y) = e^(8x) * (dy/dx) + 8e^(8x) * y
So the equation becomes:
(d/dx)(e^(8x) * y) = 5e^(8x)
Step 5: Integrate both sides of the equation
Integrating both sides of the equation with respect to x yields:
∫(d/dx)(e^(8x) * y) dx = ∫5e^(8x) dx
Integrating the left side using the chain rule, we get:
e^(8x) * y = ∫5e^(8x) dx
Solving the integral on the right side:
e^(8x) * y = (5/8) * e^(8x) + C
where C is the constant of integration.
Step 6: Solve for y
To isolate y, we divide both sides of the equation by e^(8x):
y = [(5/8) * e^(8x) + C] / e^(8x)
Simplifying the equation, we have:
y = (5/8) + Ce^(-8x)
where (5/8) is a constant and Ce^(-8x) represents the general solution of the differential equation.
Therefore, the solution to the given differential equation (dy/dx) + 8y = 5 is y = (5/8) + Ce^(-8x), where C is a constant.