23.5 Exact Equations
Sometimes you have a differential equation of the form
where Nx = My. In this happy situation, one can find a function of two variables f
and the solution to the equation is of the form
where C is a constant. This function f is called a scalar potential or potential for short.
These equations are called exact. Why does ∗ yield a solution? Say the above relation defines y as a
function of x. Then using the chain rule,
It is easy to see that if there exists a C2
with the property that fx
. This follows because My
By equality of mixed partial derivatives, you
need to have My
. In fact, if this last condition holds, then there will generally be such a potential
Why is it that if Nx = My then there exists f with the properties described?
and formally differentiating across the integral,
In general, this process of
has not been proved, but in examples, it will
be obviously true. Also, it is formally true when you think of the integral as a sort of sum and use the fact
that the derivative of a sum is the sum of the derivatives.
Example 23.5.1 Find the solutions to
You see that this is exact (2x = 2x). Then the f
. Then taking the partial derivative with respect to
it follows that
and so is suffices to let g
Then the solutions to this differential equation
where C is a constant which would be determined by some sort of an initial condition.
Example 23.5.2 In the above example, determine C if
is to be on the curve which
yields a solution to the differential equation.
You need to have 1 = C because
and so the solution in this case is sin
All of the examples of this sort of thing are similar. Exact equations are easy to solve. Physically the
solution to these equations is really a statement about the energy being constant.
Procedure 23.5.3 To solve an exact equation
do the following:
- Check to see if it really is exact by seeing if Nx = My. If it is, find a scalar potential f
such that fx = M,fy = N.
- The general solution is f =
C. Choose C to satisfy initial conditions.