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midterm_1_review (dragged) 4

midterm_1_review (dragged) 4 - nh y n h t n = t nh This...

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MA 36600 MIDTERM #1 REVIEW 5 #2. Choose a function h ( y ) according to the ordinary diFerential equation dh dy = N ( x,y ) ∂g ∂y = h ( y )= ° y ± N ( x,τ ) ∂g ∂y ( x,τ ) ² dτ. #3. Choose the function f ( x,y )= g ( x,y )+ h ( y ). Recall that the solution is f ( x,y )= C . In general M ( x,y ) dx + N ( x,y ) dy = 0 will not be an exact equation. Say that we can ±nd a (nonzero) function μ = μ ( x,y ) such that the equation μ ( x,y ) M ( x,y ) dx + μ ( x,y ) N ( x,y ) dy =0 is exact; such a function would be called an integrating factor . This is an exact equation when ∂y ³ μM ´ = ∂x ³ μN ´ . ²or example, the linear equation dy dx + p ( x ) y + g ( x )= ³ g ( x ) p ( x ) y ´ dx + ³ 1 ´ dy =0 has integrating factor μ ( x,y )=exp ±° x p ( τ ) ² . Hence the de±nitions of integrating factor in § 2.1 and § 2.6 are consistent. § 2.7: Numerical Approximations: Euler’s Method. Consider the graph of a solution y = y ( t ) to the initial value problem dy dt = G ( t,y ) ,y ( t 0 )= y 0 . We construct a sequence of points µ ( t 0 ,y 0 ) , ( t 1 ,y 1 ) ,. . . , ( t n ,y n ) ,. . . which approximates this graph as follows. Choose a step size h , and consider the recursive relations y n t n = G ( t n ,y n ) t n = h = y n +1 = y n +
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Unformatted text preview: + nh, y n ) · h t n = t + nh This is known as Euler’s Method . The smaller the step size h , the better the approximation. § 2.8: The Existence and Uniqueness Theorem. • Consider the initial value problem dy dt = G ( t,y ) , y ( t ) = y . In order to prove the Existence and Uniqueness Theorems of § 2.4, one constructs a sequence of “approximate solutions” µ φ ( t ) , φ 1 ( t ) , ..., φ n ( t ) , ... ¶ such that the limit φ ( t ) = lim n →∞ φ n ( t ) is the actual solution. To this end, de±ne the constant function φ ( t ) = y ; and de±ne recursively the sequence of functions φ n +1 ( t ) = ° t t G · τ, φ n ( τ ) ¸ dτ + y . This is known as the Method of Successive Approximations or the Method of Picard Iterations . The functions φ n ( t ) are good approximations to the actual solution φ ( t )....
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