Alaca
MATH A. 1005F
Fall 2008
1
FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS A rst order linear dierential equation is an equation of the form y + P (x)y = Q(x) ()
where P (x) and Q(x) continuous functions on a given interval. Method of solution: We are looking for an integrating factor I (x) such that when we multiply both sides of the equation (*) by I (x), left hand side of the equation d would be (Iy ). dx I (x)y + P (x)yI (x) = Q(x)I (x) d (Iy ) = QI dx Iy = y= How to nd I (x): I (x)(y + P (x)y ) = d ( I ( x) y ) . dx Q(x)I (x)dx 1 ( I x) Q(x)I (x)dx .
I ( x) y + P ( x) I ( x) y = I ( x) y + I ( x) y P ( x) I ( x) = I ( x) = P (x)dx = ln |I | = eln |I | = e |I | = e dI I P (x)dx
P (x) dx
dI dx
P (x) dx
Since we do not need most general integrating factor, we take I (x) = e
P (x) dx
.
A. Alaca
MATH 1005F
Fall 2008
2
Example: Solve the initial value problem y + y tan x = sin 2x, y (0) = 1.
A. Alaca
MATH 1005F
Fall 2008
3
Example: Solve the dierential equation y + 2xy = 2x3 . Solution: P (x) = 2x, Q(x) = 2x3 . I ( x) = e
P (x) dx
=e
2x dx
= ex .
2