Exact Differential Equations

The equation(1) is called an exact differential equation when it formed from a primitive be simple differentiation. Exact differential equations can be solved quite simply:

The functionoccurs be cause when we differentiate partially with respect toany function of why disappears, so must must put it back in when we integrate.

The functionoccurs be cause when we differentiate partially with respect toany function of why disappears, so must must put it back in when we integrate.

Because (1) is an exact differential equation, we can writeand we can findandoften by inspection and hence write down the solution of the eqation.

Test for Exact Equations

If we start from an expression u, then we can form the differential equation

Compare with (1) above to getandIt is a basic property of derivatives that is f and g are well behaved, continuous, differentiable functions, thenandSinceandthe test is to see that

Example: Isexact? If so. Solve it.

so the equation is exact.

By putting these two equal, we see thatand

The solution is

Example: Isexact?

No since