Final answer:
To find a value of n such that y = xⁿ • eˣ satisfies the equation xy' = (x - 19) · y, differentiate both sides of the equation with respect to x. Simplify and set the left and right sides equal to each other. Solve for x and find that x = -19, meaning the value of n does not matter.
Step-by-step explanation:
To find a value of n such that y = xⁿ • eˣ satisfies the equation xy' = (x - 19) · y, we need to differentiate both sides of the equation with respect to x.
Using the product rule, the left side becomes xⁿ•(eˣ) + xⁿ•(eˣ)•(1). Simplifying, we get xⁿ•eˣ + xⁿ•eˣ = 2xⁿ•eˣ.
The right side of the equation is (x - 19) • (xⁿ•eˣ). Setting the left and right sides equal to each other, we get 2xⁿ•eˣ = (x - 19) • (xⁿ•eˣ). Canceling out xⁿ•eˣ from both sides, we have 2x = x - 19.
Solving for x, we find that x = -19. Therefore, the value of n does not matter as long as x = -19.