ch02_4 - Ch 2.4: Differences Between Linear and Nonlinear...

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Ch 2.4: Differences Between Linear and Nonlinear Equations Recall that a first order ODE has the form y' = f ( t , y ), and is linear if f is linear in y, and nonlinear if f is nonlinear in y. Examples: y' = t y - e t , y' = t y 2 . In this section, we will see that first order linear and nonlinear equations differ in a number of ways, including: The theory describing existence and uniqueness of solutions, and corresponding domains, are different. Solutions to linear equations can be expressed in terms of a general solution, which is not usually the case for nonlinear equations. Linear equations have explicitly defined solutions while nonlinear equations typically do not, and nonlinear equations may or may not have implicitly defined solutions. For both types of equations, numerical and graphical construction of solutions are important.
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Theorem 2.4.1 Consider the linear first order initial value problem: If the functions p and g are continuous on an open interval ( α , β ) containing the point t = t 0 , then there exists a unique solution y = φ ( t ) that satisfies the IVP for each t in ( , ). Proof outline: Use Ch 2.1 discussion and results: 0 ) 0 ( ), ( ) ( y y t g y t p dt dy = = + = + = t t ds s p t t e t t y dt t g t y 0 0 ) ( 0 ) ( where , ) ( ) ( ) ( μ
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Consider the nonlinear first order initial value problem: Suppose f and f / y are continuous on some open rectangle ( t , y ) ( α , β ) x ( γ , δ ) containing the point ( t 0 , y 0 ). Then in some interval ( t 0 - h , t 0 + h ) ( , ) there exists a unique solution y = φ ( t ) that satisfies the IVP. Proof discussion: Since there is no general formula for the solution of arbitrary nonlinear first order IVPs, this proof is difficult, and is beyond the scope of this course. It turns out that conditions stated in Thm 2.4.2 are sufficient but not necessary to guarantee existence of a solution, and continuity of f ensures existence but not uniqueness of . 0 ) 0 ( ), dy = =
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This note was uploaded on 04/11/2011 for the course MECHANICAL MAT 219 taught by Professor Benjaminwalter during the Fall '10 term at Middle East Technical University.

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ch02_4 - Ch 2.4: Differences Between Linear and Nonlinear...

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