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#### karush

##### Well-known member

- Jan 31, 2012

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- Thread starter karush
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- Jan 31, 2012

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- Jan 29, 2012

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Here the problem just asks you to "State where in the ty-plane the hypotheses of theorem 2.4.2 are satisfied". The hypotheses of theorem 2.4.2 are that the function f(t,y) in the differential equation y'= f(t,y) be

\frac{t- y}{2t+ 5y}$ continuous?"

You should know that a rational function, such as this, is continuous as long as the denominator is not 0. So we want to find (t, y) such that $2t+ 5y\ne 0$. The simplest way to do that is to say where it**is** 0! 2t+ 5y= 0 is a straight line in the ty-plane. That is equivalent to the line y= -(2/5)t, a line through the origin with slope -2/5. The hypotheses of theorem 2.4.2 are satisfied every where EXCEPT on that line.

You should know that a rational function, such as this, is continuous as long as the denominator is not 0. So we want to find (t, y) such that $2t+ 5y\ne 0$. The simplest way to do that is to say where it