Solve the given differential equation by separation of variables. x dy/dx = 6y.

Solution:

Given x dy/dx = 6y

The given a differential equation of this form:

dydx = h(x)g(y)

Then we can separate the variables x and y in order to solve the equation:

dydx = h(x)g(y) ⇒ g(y)dy = h(x)dx ⇒ ∫g(y)dy = ∫h(x)dx

Rearrange as follows:

1/x dx = 1/6y dy

Integrate on both sides

lnx + c = ln6y + c

Apply anti-log on both sides

x = 6y + c

Hence, the solution is x = 6y + c

Solve the given differential equation by separation of variables. x dy/dx = 6y.

Summary:

The given differential equation by separation of variables. x dy/dx = 6y is x = 6y + c.