Separable Equations

The separation method

Worked forms

Modeling with separation

The scope of separation

A differential equation is separable when the right side can be written in which form?

To solve dy/dx = x y, which separated form results?

Separation of dy/dx = x/y gives y squared over 2 = x squared over 2 + C. If y cannot be conveniently isolated, what is the status of this relation?

A well-stirred tank's salt amount y(t) follows dy/dt = 20 - y/50. As t approaches infinity, y(t) approaches what value?

Separating dy/dx = y(1 - y) by dividing by y(1 - y) loses which solutions?

Separating dy/dx = ky gives dy/y = k dx. Integrating the left side gives ln|y|. As a single expression, what is it? Answer like ln|y|.

Newton's cooling dT/dt = k(T - T_env) separates to dT/(T - T_env) = k dt. What solution form results?

For dy/dx = y(1 - y), the values y = 0 and y = 1 make the right side zero, giving constant solutions called what? Answer with one word.

The mixing equation dy/dt = 20 - y/50 rewrites as dy/(1000 - y) = dt/50 after factoring. As a fraction, what is the coefficient of y in the denominator's form 1000 - y? Answer with a number.

Dividing by h(y) during separation is invalid where h(y) = 0, possibly losing solutions. The constant solutions so lost are recovered by what? Answer with two words.