Differential Equations

Question 1

What is a function f(x,y), Mdx + Ndy = 0 said to be exact?

Answer 1

When ∂M/∂y = ∂N/∂x

Question 2

How do you solve an exact equation?

Answer 2

1. Confirm it is exact
2. An exact differential equation has a solution of c = f(x,y), such that
   ∂f/∂x = M(x,y) and ∂f/∂y = N(x,y). To find this function, integrate M(x,y) with respect to ‘x’ and write ‘+ h(y)’ instead of the usual ‘+ c’.
3. Then, differentiate this equation partially with respect to y: ∂f(x,y)/∂y, writing h’(y) in place of h(y).
4. Compare this partial derivative with N(x,y) to find your value of h’(y).
5. Integrate this value of h’(y) found with respect to ‘y’.
6. Place your answer in the f(x,y) you got from integrating M(x,y), and equate it to c.

Question 3

What is the order of a differential equation?

Answer 3

Whether it has been differentiated once or twice or thrice:

d²y/dx² is a second order diff cause it has been differentiated twice.

Question 4

What is the degree of a differential equation?

Answer 4

What power has it been raised to? One, two, etc:

(dy/dx)² is a diff equation of degree 2