Exact Differential Equation Math Is Fun

The goal of this section is to go backward. Dy dx p x y q x yn where n is any real number but not 0 or 1 when n 0 the equation can be solved as a first order linear differential equation. G x x 3 2x c equation 2 now we can replace the g x in equation 2 in equation 1.

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And the general solution is of the form.

Exact differential equation math is fun. I x y c. The equation f x y c gives the family of integral curves that is the solutions of the differential equation. When the population is 1000 the rate of change dn dt is then 1000 0 01 10 new rabbits per week. A bernoulli equation has this form.

Let us imagine the growth rate r is 0 01 new rabbits per week for every current rabbit. I x y 2y 3 x 2 y 3y x 3 2x c. Think of dn dt as how much the population changes as time changes for any moment in time. For some function f x y then it is automatically of the form df 0 so the general solution is immediately given by f x y c.

Therefore if a differential equation has the form. Observe that they are first order when there is only dy dx not d2y dx2 or d3y dx3 etc. 2xy g x 3x 2 2xy 2. X 6 6 2 18.

Dy dx p x y q x where p x and q x are functions of x. So the general solution of the differential equation is. The next type of first order differential equations that we ll be looking at is exact differential equations. Before we get into the full details behind solving exact differential equations it s probably best to work an example that will help to show us just what an exact differential equation is.

A differential equation of type is called an exact differential equation if there exists a function of two variables with continuous partial derivatives such that the general solution of an exact equation is given by. If you have an equation like this then you can read more on solution of first order linear differential equations. With a 9 b 6 and c 1. We can write this equation in differential form as f x x y dx f y x y dy 0 now divide by dx we are not pretending to be rigorous here to get f x x y f y x y dfrac dy dx 0 which is a first order differential equation.

X 6 6 2 4 9 1 2 9. X b b2 4ac 2a. G x 3x 2 2. 0 2xy 0 g x 3x 2 2xy 2.

This does not factor easily so we use the quadratic equation formula. X 6 36 36 18. In this case is called an exact. Non linear differential equations are often harder to solve and therefore commonly approximated by linear differential equations to find an easier solution.

9r 2 6r 1 0. X 1 2 3.