Question

Find the general solution of y' = y/x - e(⁷ʸ/ˣ).

A) y(x) = x(e(⁷/ˣ)⁺C)
B) y(x) = x²(e(⁷/ˣ)⁺C)
C) y(x) = xe(⁷/ˣ) + C
D) y(x) = x²e(⁷/ˣ) + C

Answer 1

None of the provided options A) through D) directly match the general solution for the given differential equation after applying the integrating factor technique. An error or typo in the question or options may be causing this inconsistency.

Step-by-step explanation:

To find the general solution of the differential equation
`y' = y/x - e(7y/x),` we recognize this as a first-order linear differential equation in standard form. We can solve it using an integrating factor. The goal is to manipulate the equation such that the left side becomes the derivative of a product of a function
`u(x)`, which is the integrating factor, and the solution function First, recognize the standard form for the integrating factor technique, which is
`y' + P(x)y = Q(x)`. In our equation,
`P(x) = -1/x, and Q(x) = -e7`, which suggests that the integrating factor

`u(x) is e∫ P(x)dx = e-∫ (1/x)dx = e-ln|x| = 1/x`.

Multiplying the entire differential equation by this integrating factor, we end up with
`(1/x)y' - (1/x2)y = -(1/x)e(7y/x)`. Now, the left-hand side is the derivative of
`y/x.` Integrating both sides with respect to x gives us

`y/x = ∫ -(1/x)e(7y/x)dx + C`, where C is the integration constant. The term e
`(7y/x)` complicates the integration, unless we treat
`y/x` itself as a new variable, which leads to
`z = y/x`. By substituting back into our equation, we end up with

`z = -e7zdx`, and then integrating z we find that z equals the expression in terms of x and C. After performing the integration and solving for y, none of the provided solutions A) through D) directly match. Therefore, we cannot confidently say which of the given options is correct. If there is a misunderstanding or typo in the original or provided options, please clarify so we can attempt the problem again.