Lesson 6a - Bernoulli Differential Equations

Wecannowapplythelinearstrategyon this ode

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Unformatted text preview: x)u b( x) 1 c u ' (1 c) a ( x)u (1 c)b( x) Notice that this is now linear! We can now apply the linear strategy on this ODE. Strategy for Solving Bernoulli Equations y ' a ( x) y b( x) y c Step 1: If c = 0 or 1, simplify the original ODE and solve it using the linear strategy. 1 1 c yu u '(1 c)a ( x)u (1 c)b( x) Step 2: If c ≠ 0 or 1, let , and solve using the linear strategy. (1 c) I ( x)b( x)dx I ( x) e (Reminder: , then find . This makes a (1 c ) a ( x )dx 1 I ( x) u (1 c) I ( x)b( x)dx solution for u as: ) u y1c Step 3: Replace , and solve for y accordingly. Example 1 Determine the general solution to xy ' y xy ' y 3 x y 3 2 3 2 xy ' y 3 x y xx x 1 y ' y 3x 2 y 2 x y ' a ( x) y b( x) y c 1 a( x) b( x ) 3 x 2 c 2 x y u 1 1 c u '(1 c)a ( x)u (1 c)b( x) y u 1 1 u ' (1) u (1)3 x 2 x 3x 3 y 2 I ( x) e 1 dx x I ( x) 1 I ( x)b( x)dx (3x 2 )dx x 3xdx 1 x 32 x c 2 1 I ( x)b( x)dx I ( x) 1 y 32 33 u x( x c) x cx 2 2 1 33 x cx y2 u Example 2 Determine the general solution to the IV...
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