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] , so we can integrate it. But how do we find I(x)? We want I(x)(y + P(x)y) = (I(x)y) Expand I(x)y + I(x)P(x)y = I (x)y + I(x)y (used product rule on RHS) So I(x)P(x) = I (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] , which in this case is (xy) New equation is (xy) = 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 + xy = 1 (x > 0), where y(1) = 2. [Hint: If there is a number or a variable in front of the y , you need to divide first.] Example 4 Find the solution of y + 2xy = 1.