Ordinary Diff Eq Exam Review Solutions 23

Ordinary Diff Eq Exam Review Solutions 23 - 1 Let are...

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Unformatted text preview: 1 Solutions 25 27. Let are denote the volume of ﬂuid in the tank at time . Initially, there gal of brine. For each minute that passes there is a net decrease of gal of brine. Thus gal. 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 Put in standard form, this equation becomes The coeﬃcient function is , , and the integrating factor is . Multiplying the standard form equation by the integrating factor gives Integrating and simplifying gives condition implies and hence Of course, this formula is valid for no ﬂuid and hence no salt in the tank. 28. input rate: input rate output rate: g L minutes there is g min L g min L output rate, it follows that output rate Since input rate value problem L min . After The initial so g . min satisﬁes the initial Put in standard form, this equation becomes . The coeﬃcient function is , , and the integrating factor is . Multiplying the standard form equation by the integrating factor gives . Integrating and simplifying gives The initial condition implies so After minutes we have g of salt. The concentration is thus g/L ...
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