Question

Find the general solution for xy' - 3y = 4x.

a) y = (4/x) + Ce³/x
b) y = (4/x) + C/x³
c) y = (4/x) + C/x²
d) y = (4/x) + Cx³

Answer 1

The correct general solution to the differential equation
`xy' - 3y = 4x is b)`
`y = (4/x) + C/x^3`, found using the method of integrating factors and integrating both sides of the rearranged equation.

Step-by-step explanation:

The student is tasked with finding the general solution for the first-order linear differential equation
`xy' - 3y = 4x`. To solve this, we can use the method of integrating factors. First, we rewrite the equation in the standard form
`y' + P(x)y = Q(x)`, where
`P(x) = -3/x and Q(x) = 4`.

The integrating factor,
`μ(x)`, is found using

`µ(x) = e∫P(x)dx so µ(x) = e∫(-3/x)dx = x-3`. Multiplying both sides of the equation by
`µ(x) gives x-3xy' - 3x-3y = 4x-3x`. Uponsimplification, we get
`(x-2y)' = 4x-3`. Integrating both sides gives
`x-2y = -2x-2 + C` where C is the constant of integration. Then multiply by
`x2` to find the general solution
`y = (4/x) + C/x3`. Therefore, the correct answer from the given options is b)
`y = (4/x) + C/x3`.