First Order Differential Equations:

dy/dx = f(x,y) or y’ = f(x,y)

Special Types of First Order Equations:

  1. Separable: dy/dx = g(x) h(x)

Solution Method: Separate and Integrate

II. Linear: a (x)dy/dx + a (x)y = g(x)

Standard form: dy/dx + P(x)y = Q(x)

Integrating Factor: e

Solution Method: Multiply thru by the IF and then integrate (look for the Product Rule)

First Order IVP:

Solve dy/dx = f(x,y) subject to y(x ) = y

Unique solution in a rectangle R containing (x ,y ) if:

    1. f(x,y) is continuous throughout R.
    2. f/ y is continuous throughout R.

III. Homogeneous: dy/dx = g(y/x)

Solution Method: Let v = y/x or y = vx and substitute and solve the resulting separable equation

IV. Exact: M(x,y)dx + N(x,y)dy = 0

M/ y = N/ x

Solution Method: Find F(x,y) such that F/ x = M and F/ y = N and the solution (g) is contained in the equation F(x,y) = c

V. Bernoulli: Let w = y as long as n {0,1}

dy/dx + P(x)y = Q(x)y