Identify the type of differential equation, and solve it with the initial condition H(0) = 0.
There are many important differential equations for which separation of variables fails, but which can be solved with other techniques. An important category involves Newton's law of cooling when the ambient temperature is changing. Consider, in particular, the case where A(t) = eβt. Assume that the constant α is 1.0, so that the differential equation is dH/dt = -H + A(t). It is impossible to separate variables in this equation.