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A mathematical model for the rate at which a drug disseminates into the bloodstream is given by dx =r - kx, dt where r and k are positive constants. The function x(t) describes the concentration of the drug in the bloodstream at time t. (a) Since the DE is autonomous, use the phase portrait concept of Section 2.1 to find the limiting value of x(t) as t → ∞. lim x(t) = t- 00 (b) Solve the DE subject to x(0) = 0. x(t) = Sketch the graph of x(t) and verify your prediction in part (a). k k At what time is the concentration one-half this limiting value?

A mathematical model for the rate at which a drug disseminates into the bloodstream is given by dx =r - kx, dt where r and k are positive constants. The function x(t) describes the concentration of the drug in the bloodstream at time t. (a) Since the DE is autonomous, use the phase portrait concept of Section 2.1 to find the limiting value of x(t) as t → ∞. lim x(t) = t- 00 (b) Solve the DE subject to x(0) = 0. x(t) = Sketch the graph of x(t) and verify your prediction in part (a). k k At what time is the concentration one-half this limiting value?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A mathematical model for the rate at which a drug disseminates into the bloodstream is given by dx =r - kx, dt where r and k are positive constants. The function x(t) describes the concentration of the drug in the bloodstream at time t. (a) Since the DE is autonomous, use the phase portrait concept of Section 2.1 to find the limiting value of x(t) as t → ∞. lim x(t) = (b) Solve the DE subject to x(0) = 0. x(t) = Sketch the graph of x(t) and verify your prediction in part (a). r k r k X X t At what time is the concentration one-half this limiting value?
Transcribed Image Text:A mathematical model for the rate at which a drug disseminates into the bloodstream is given by dx =r - kx, dt where r and k are positive constants. The function x(t) describes the concentration of the drug in the bloodstream at time t. (a) Since the DE is autonomous, use the phase portrait concept of Section 2.1 to find the limiting value of x(t) as t → ∞. lim x(t) = (b) Solve the DE subject to x(0) = 0. x(t) = Sketch the graph of x(t) and verify your prediction in part (a). r k r k X X t At what time is the concentration one-half this limiting value?
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