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notes 2-8 - SECTION 2.8 GETTING CLOSE TO A SOLUTION WHEN...

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SECTION 2.8 GETTING CLOSE TO A SOLUTION WHEN YOU CAN’T FIND ONE Suppose we have a first order initial value problem y 0 ( t ) = f ( t, y ( t )) , y ( t 0 ) = y 0 but the stupid thing is not linear nor exact nor separable, and we can’t find an integrating factor. BUT WE NEED VALUES FOR y ( t )!!! What do we do? We could use Euler’s Method to produce numerical approximations. The trouble is that this doesn’t provide or even give any good idea about a formula for a solution y ( t ). There are several ideas which can in fact provide more information. IDEA NUMBER ONE. Change the initial value problem y 0 ( t ) = f ( t, y ( t )) , y ( t 0 ) = y 0 into the initial value problem y 0 1 ( s ) = g ( s, y 1 ( s )) , y 1 (0) = 0 by putting y 1 ( s ) = y ( s + t 0 ) - y 0 . This is a technical trick that allows us to think only about the initial condition y (0) = 0 and we say/do no more about it. IDEA NUMBER TWO. Change the initial value problem y 0 ( s ) = g ( s, y ( s )) , y (0) = 0 into an integral equation. Suppose φ ( s ) is a solution of the initial value problem y 0 ( s ) = g ( s, y ( s )) , y (0) = 0. Then φ 0 ( s ) = g ( s, φ ( s )) . Integrate both sides from s = 0 to
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