Answered: A one-parameter family of solutions of…

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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Question

A one-parameter family of solutions of the DE P' = P(1 − P) is given below. P = c₁e^t/1 + c₁e^t.

Does any solution curve pass through the point (0, 9)? Through the point (0, 1)? (If yes, give the corresponding value of c₁. If not, write DNE.)

Expert Solution

For the point (0,9)

We are given that

$P = \frac{c_{1} e^{t}}{1 + c_{1} e^{t}}$ ----(1)

Now, we are given the point (0,9)

$\Rightarrow 9 = \frac{c_{1} e^{0}}{1 + c_{1} e^{0}} \Rightarrow 9 = \frac{c_{1}}{1 + c_{1}} \Rightarrow 9 + 9 c_{1} = c_{1} \Rightarrow c_{1} = \frac{- 9}{8}$

Yes the solution curve passes through (0,9).The corresponding value of c₁ is $\frac{- 9}{8}$

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