Question

Solve 4xy′−4y=x−9,y(1)=4 Identify the integrating factor, r(x). (Note that the equation is not in standard form for 1 st-order linear. It must be in order to identify p(x) r(x)= Find the general solution. y(x)= Note: Use C for the arbitrary constant. Solve the initial value problem y(1)=4. y(x)=

Answer 1

Integrating both sides with respect to x, we obtain
`e^(-2ln|x|) * y = ∫(x - 9e^(-2ln|x|))/`4x dx. By solving the integral and simplifying, we find
`y(x) = (x^2 - 5x + 4) / 4.`

Explanation:

To solve the differential equation 4xy′ − 4y = x − 9, we first identify the integrating factor, which in this case is r(x) = e^(-2ln|x|). Rearranging the equation into standard form allows us to rewrite it as
`d/dx [e^(-2ln|x|) * y] = (x - 9e^(-2ln|x|))/4x.` Integrating both sides with respect to x, we obtain
`e^(-2ln|x|) * y = ∫(x - 9e^(-2ln|x|))/`4x dx. By solving the integral and simplifying, we find
`y(x) = (x^2 - 5x + 4) / 4.`

The integrating factor, crucial to transforming the equation into standard form, involves recognizing the form of the differential equation and using it to identify the appropriate factor. The subsequent integration of both sides brings the equation into a form that allows for straightforward manipulation to solve for y(x) in terms of x. Applying the initial condition y(1) = 4 yields the specific solution provided earlier.

This process showcases the utilization of integrating factors to manipulate non-standard differential equations into forms conducive to solving for the general solution and subsequently determining the particular solution based on the given initial condition.