Ordinary Diff Eq Exam Review Solutions 24

# Ordinary Diff Eq Exam Review Solutions 24 - 26 1 Solutions...

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Unformatted text preview: 26 1 Solutions 29. Let denote the volume of ﬂuid in the container at time . Initially, there are L. For each minute that passes there is a net gain of L of ﬂuid. So . The container overﬂows when or minutes. L g g . input rate: input rate min L min L g g output rate: output rate min L min Since input rate output rate, it follows that satisﬁes the initial value problem Simplifying and putting in standard form gives the equation , , and the The coeﬃcient function is integrating factor is . Multiplying the standard form equation by the integrating factor gives . Integrating and simplifying gives , where is a constant. The initial condition implies so At the time the container overﬂows we have g of salt. 30. Let denote the volume of ﬂuid in the tank at time . Initially, there are gallons of ﬂuid. For each minute that goes by there is a net increase of gallons. It follows that . The tank will overﬂow when . Solving gives . Thus minutes. Next we ﬁnd : gal lbs lbs input rate: input rate . min gal min gal lbs lbs output rate: output rate . min gal min Since input rate output rate, it follows that satisﬁes the initial value problem Putting this equation in standard form gives The coeﬃcient function is integrating factor is , . Thus , and the . Integrating ...
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