Final answer:
The differential equation dx/dt = 9xt is separable but not linear, as the variables can be separated and integrated to find the solution for x in terms of t. To solve such equations, known values must be listed, and variables should be rearranged appropriately.
Step-by-step explanation:
The student is asking about the nature of the differential equation dx/dt = 9xt. This equation is separable because we can separate the variables x and t to opposite sides of the equation by dividing both sides by x and then multiplying both sides by dt, resulting in dx/x = 9t dt. After the separation, we integrate both sides to find a function for x in terms of t. This equation is not linear since it contains the product of the two variables x and t.
Mathematically, separable equations can be expressed as dx/dt = f(x)g(t), where the function can be split into two separate functions of x and t. Integrating these functions separately gives a solution to the original differential equation.
It is important to note that when dealing with differential equations, it is crucial to make a complete list of the knowns and to rearrange the equation to solve for the unknown, in this case the function x(t). The physical significance of derivatives and their relation to real-world problems, such as acceleration and velocity in physics, should also be considered.