Question

Solve xy' = 5y – 3х, y(1) = 8.

(a) Identify the integrating factor, μ(π).
μ(x) =
(b) Find the general solution.
y(x) =
Note: Use C for the arbitrary constant.
(c) Solve the initial value problem y(1) = 8. y(x)
=

Answer 1

To solve the given first order linear differential equation, we must first identify the integrating factor, μ(x), and then use it to find the general solution y(x). The initial condition y(1) = 8 helps determine the particular solution by finding the constant C.

Step-by-step explanation:

The differential equation given is xy' = 5y - 3x, which is a first order linear differential equation that can be solved using an integrating factor. To solve for the integrating factor, μ(x), the equation must be in the standard form, which is y' + P(x)y = Q(x). However, the given equation needs to be rearranged to this form by dividing every term by x, which gives y' - (5/x)y = -3.

An integrating factor is generally given by μ(x) = e∫ P(x)dx. For this equation, P(x) = -5/x, so the integrating factor will be μ(x) = e∫ -5/x dx = x-5. To find the general solution, y(x), we multiply both sides of the standard form by μ(x) and integrate.

The result is then used to find the particular solution that meets the initial condition y(1) = 8. After solving the specific equation with the given boundary condition, we can find the value of the arbitrary constant, C.