course materials - MA 36600 MIDTERM #2 REVIEW Chapter 3...

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Unformatted text preview: MA 36600 MIDTERM #2 REVIEW Chapter 3 3.1: Homogeneous Equations with Constant Coefficients. An ordinary differential equation in the form d 2 y dt 2 = G t, y, dy dt is called a second order differential equation . If we have initial conditions in the form y ( t ) = y and dy dt ( t ) = y 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 differential equation nonlinear . We will consider equations in the form a ( t ) y 00 + b ( t ) y + c ( t ) y = f ( t ) . The functions a ( t ), b ( t ), and c ( t ) are called coefficients . We say that the differential 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 00 + by + cy = 0. We guess that a solution is y = e rt for some constant r ; this yields the 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 , then the solution to the initial value problem ay 00 + by + cy = 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 00 + b ( t ) y + c ( t ) y = f ( t ); y ( t ) = y , y ( t ) = y . Say that we have an interval I = t R < t < 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 ) 6 = 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 differential equations. 1 2 MA 36600 MIDTERM #2 REVIEW Say that y 1 = y 1 ( t ) and y 2 = y 2 ( t ) are solutions to the homogeneous equation a ( t ) y 00 + b ( t ) y + c ( t ) y = 0 . The Principle of Superposition states that the linear combination y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) is also a solution for any constants c 1 and c 2 . We say that { y 1 ,y 2 } is a Fundamental Set of Solutions for the differential equation if this is the general solution. Let y 1 = y 1 ( t ) and y 2 = y 2 ( t ) be any two functions. They are said to be linearly independent if the equation c 1 y 1 ( t ) + c 2 y 2 ( t ) = 0 for all t has the unique solution c 1 = c 2 = 0. Otherwise, we say y 1 and y 2 are linearly dependent . Let y 1 = y 1 ( t ) and y 2 = y 2 ( t ) be any two differentiable functions. The Wronskian of y 1 and y 2 is W ( y 1 ,y 2 ) ( t ) = y 1 ( t ) y 2 ( t )- y 1 ( t ) y 2 ( t ) = det y 1 ( t ) y 2 ( t ) y 1 ( t ) y 2 ( t ) ....
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This note was uploaded on 05/24/2011 for the course MA 36600 taught by Professor Ma during the Spring '09 term at Purdue University-West Lafayette.

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course materials - MA 36600 MIDTERM #2 REVIEW Chapter 3...

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