Linear Differential Equations AP CALCULUS BC

First-Order Differential Equations  A first-order linear differential equation can be put into the form where P and Q are continuous functions on a given interval.  These types of equations occur frequently in various sciences.  These equations are not separable – we cannot rewrite it as f(x)g(y). So what do we do?

Integrating Factor  We can solve any first-order linear differential equation by multiplying both sides by the integrating factor, I(x).  Our goal is to get the left side to equal [I(x) y] ʹ, so we can integrate it.  But how do we find I(x)?  We want I(x)(y ʹ + P(x)y) = (I(x)y) ʹ  Expand  I(x)y ʹ + I(x)P(x)y = I ʹ (x)y + I(x)y ʹ (used product rule on RHS)  So I(x)P(x) = I ʹ (x)

Integrating Factor (cont.)  This is a separable differential equation if we rewrite it as  Therefore,  Integrate to get   And finally! 

Example 1 Find the general solution to the differential equation 1)First find I(x)  2)Multiply both sides by the integrating factor  3)Replace the left side with [I(x) y] ʹ, which in this case is (xy) ʹ New equation is (xy) ʹ = 2x 4)Integrate both sides  5)Solve for y 

Example 2 Solve the differential equation

Example 3 Find the solution of the initial value problem x 2 y ʹ + xy = 1 (x > 0), where y(1) = 2. [Hint: If there is a number or a variable in front of the y ʹ, you need to divide first.]

Example 4 Find the solution of y ʹ + 2xy = 1.