Final answer:
The general solution of the differential equation dy/dx = 10√(xy) is found by separating variables, integrating both sides, and then squaring the result, yielding y = (10√x + C/2)^2 where C is the constant of integration.
Step-by-step explanation:
To find the general solution of the differential equation dy/dx = 10√(xy), we must first separate variables. We can do this by dividing both sides of the equation by √(y) and then multiplying both sides by dx to isolate terms involving y on one side and terms involving x on the other.
After separation, we integrate both sides of the equation. For the left side, the integral of 1/√y with respect to y is 2√y. For the right side, we have the integral of 10√x with respect to x, which is 20√x. Integrating both sides gives us
2√y = 20√x + C,
where C is the constant of integration. Finally, to obtain the explicit solution for y, we square both sides:
y = (10√x + C/2)^2.
This is the general solution for the differential equation given that x and y are positive.