Final answer:
To solve the initial value problem y' = (y/x) * e^x, use separation of variables and integration. The solution is xy = C, where C is a constant.
Step-by-step explanation:
To solve the initial value problem y' = (y/x) * e^x, we can use separation of variables. Rearranging the equation, we have y' - (y/x) = 0. Next, multiply both sides of the equation by x to get xy' - y = 0. Now, we can rewrite the equation as (xy)' = 0. Integrating both sides, we get xy = C, where C is the constant of integration.
Now, to find the value of C, we can use the initial condition. Let's say the initial condition is y(1) = 2. Plugging this into the equation xy = C, we have 1(2) = C, which gives us C = 2.
So, the solution to the initial value problem y' = (y/x) * e^x with the initial condition y(1) = 2 is xy = 2.