Question

Suppose we have the autonomous ODE:

y′ = y^(2)(y − 2)

Answer 1

To solve the given autonomous ODE y' = y^2(y - 2), we can separate the variables and integrate.

Step-by-step explanation:

The given equation is a first-order autonomous ordinary differential equation (ODE).

To solve this ODE, we can separate the variables and integrate.

Divide both sides by y^2(y - 2) and rewrite the equation as:

dy/(y^2(y - 2)) = dx

Integrate both sides:

∫[(1/y^2(y - 2))] dy = ∫dx

Now, solve the integral on the left side:

Let u = y - 2, then du = dy

∫[-1/[(u + 2)u^2]] du = x + C

After evaluating the integral, we get the solution:

ln|y - 2| + 1/(y - 2) = x + C