= F (x, y). If F (x, y) =

,

dx h(y) it is called a separable equation and can be solved by the method of separation of variables: h(y) dy = g(x) dx.

• A ﬁrst order diﬀerential equation has the form

dy

• A ﬁrst order linear diﬀerential equation is of the form

+ P (x)y = Q(x). To dx solve the equation, we multiply both sides of the equation by the integrating factor

I(x) := e

P (x) dx

. The equation becomes

d

I(x)y = I(x)Q(x) dx or

I(x)y =

I(x)Q(x) dx.

• A second order homogeneous diﬀerential equation with constant coeﬃcients has the form ay + by + cy = 0 (a = 0). The characteristic polynomial of the equation is ar2 + br + c. The general solution of the equation is

(a) y = C1 er1 x + C2 er2 x if ar2 + br + c has two distinct real roots r1 and r2 ;

(b) y = C1 erx + C2 xerx if ar2 + br + c has a double real root r;

(c) y = C1 eαx sin (βx) + C2 eαx cos (βx) if ar2 + br + c has two distinct complex roots r1 = α + βi and r2 = α − βi.

• The general solution of the nonhomogeneous second order linear diﬀerential equation ay + by + cy = G(x) can be written as y(x) = yp (x) + yc (x) where yp is a particular solution of the equation and yc is the general solution of the complementary equation ay + by + cy = 0. A particular solution can be found by using the method of variation of parameters.

• The method of variation of parameters always works. Let y1 and y2 be two linearly

independent…