Exact Equations
Form: M(x,y) dx + N(x,y) dy = 0
1. Check if the equation is exact:
if dM/dy = dN/dx, then the equation is exact.
note that M is simply the partial derivative with respect to x of a function f(x,y), and N is the partial derivative of f with respect to y.
2. If it is exact:
- find f(x,y) by taking the integral of M dx, or N dy.
- the constant should be written as h(y) or h(x) respectively if you chose to integrate M dx or N dy.
- find the partial derivative of the function f(x,y) you just found with respect of the other variable (y or x respectively), you should end up with a h'(y) or h'(x) respectively.
- make it equal to N or M respectively, terms should cancel, so that...
- you can now integrate both sides to isolate the constant.
- f(x,y) = [the partial derivative of x or y] + [h(y) or h(x)] = constant
3. If it is not exact:
-
if dM/dy = dN/dx, then the equation is exact.
note that M is simply the partial derivative with respect to x of a function f(x,y), and N is the partial derivative of f with respect to y.
2. If it is exact:
- find f(x,y) by taking the integral of M dx, or N dy.
- the constant should be written as h(y) or h(x) respectively if you chose to integrate M dx or N dy.
- find the partial derivative of the function f(x,y) you just found with respect of the other variable (y or x respectively), you should end up with a h'(y) or h'(x) respectively.
- make it equal to N or M respectively, terms should cancel, so that...
- you can now integrate both sides to isolate the constant.
- f(x,y) = [the partial derivative of x or y] + [h(y) or h(x)] = constant
3. If it is not exact:
-