midterm_2_review (dragged)

# midterm_2_review (dragged) - °° b y ° c y = 0 y t = y y...

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MA 36600 MIDTERM #2 REVIEW Chapter 3 § 3.1: Homogeneous Equations with Constant Coeﬃcients. An ordinary diferential equation in the Form d 2 y dt 2 = G ° t, y, dy dt ± is called a second order diferential equation . IF we have initial conditions in the Form y ( t 0 )= y 0 and dy dt ( t 0 )= y ° 0 we call this system an initial value problem . Such an equation is called a linear equation iF G ( t, y 1 ,y 2 )= g ( t ) q ( t ) y 1 p ( t ) y 2 = d 2 y dt 2 + p ( t ) dy dt + q ( t ) y = g ( t ) . Otherwise, we call the diferential equation nonlinear . We will consider equations in the Form a ( t ) y °° + b ( t ) y ° + c ( t ) y = f ( t ) . The Functions a ( t ), b ( t ), and c ( t ) are called coeﬃcients . We say that the diferential equation is homogeneous iF f ( t ) = 0 For all t ; otherwise, it is said to be a nonhomogeneous equation . IF we have a constant coeﬃcient homogeneous equation, it is in the Form ay °° + by ° + cy = 0. We guess that a solution is y = e rt For some constant r ;th isy ie ldsthe characteristic equation ar 2 + br + c =0 . IF the characteristic equation ar 2 + br + c = 0 has two distinct real roots r 1 and r 2
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Unformatted text preview: °° + b y ° + c y = 0 , y ( t ) = y , y ° ( t ) = y ° ; is the Function y ( t ) = c 1 e r 1 t + c 2 e r 2 t in terms oF the constants c 1 = y ° − r 2 y r 1 − r 2 e − r 1 t and c 2 = y ° − r 1 y r 2 − r 1 e − r 2 t . § 3.2: Solutions of Linear Homogeneous Equations; the Wronskian. • Consider the initial value problem a ( t ) y °° + b ( t ) y ° + c ( t ) y = f ( t ); y ( t ) = y , y ° ( t ) = y ° . Say that we have an interval I = ² t ∈ R ³ ³ α < t < β ´ such that i. f ( t ) is continuous on I , ii. both b ( t ) and c ( t ) are continuous on I , iii. a ( t ) is continuous yet a ( t ) ° = 0 on I , and iv. t ∈ I . Then there exists a unique solution y = y ( t ) to the initial value problem. This is the Existence and Uniqueness Theorem For linear second order diferential equations. 1...
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