Find an explicit solution of the given initial-value problem The image presents a first-order differential equation and an initial condition. The differential equation is:\[ \frac{dy}{dx} = \frac{y^2 - 1}{x^2 - 1} \]There is also an initial condition given:\[ y(7) = 7 \]The goal is to find the function \( y \) that satisfies both the differential equation and the initial condition, and there is an empty box labeled with \( y = \) to fill in the solution. There are no graphs or diagrams associated with this image.

Answered: Image /qna-images/answer/60e533ab-084f-49cb-9cc4-05d5a1fa61fb.jpg

Answered: dy y? – 1 y(7) = 7 1 dx - y =

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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q5
The evolution of a population with constant migration rate M is described by the initial value problem dP = kP + M. dt P(0) = Po. (a) Solve this initial value problem; assume k is constant. (b) Examine the solution P(t) and determine the relation between the constants k and M that will result in P() remaining constant in time and equal to Po- Explain, on physical grounds, why the two constants k and M must have opposite signs to achieve this constant equilibrium solution for P(t).
11
This is the fourth part of a four-part problem. If the given solutions ÿ₁ (t) = 2e³t 8e 3est 20e-t form a fundamental set (i.e., linearly independent set) of solutions for the initial value problem ÿ' = 2 (t) = ÿ, 7(0) = = 2e³8e- 3e320e-t 8e³+2e7 12e³t+5e7 [17], impose the given initial condition and find the unique solution to the initial value problem. If the given solutions do not form a fundamental set, enter NONE in all of the answer blanks. 8e³+2e- -O [12³ + 5e-² g(t)=
A salmon population is modelled by the DTDS X+1 = 0.3 + x, • (a) ( Write down the updating function f (x) of this DTDS. (b) ( Suppose we know that X10 = 0. 5. What is the value of X11? • (c) SWrite down the equation that we need to solve in order to find the fixed points. Then solve the equation algebraically (that is, without using graphs). • (d) point. (No marks will be given if you use other methods.) Use the Stability Theorem to determine the stability of each fixed
The initial value problem modeling the number of fish in a stocked pond after t years is dy = = .08y — b, y(0) = 400. b is greater than zero and represents the harvesting rate. dt a) Find the value of b which causes a non-zero steady state solution. b) Solve the initial value problem. c) What is the steady state solution if b is smaller than the value found in part a of this problem? d) What is the steady state solution if b is larger than the value found in part a of this problem?
Justify and analyze the model proposed in equation (2), covering in particular the following issues: an analysis of the validity of equation (2) as a model for the fish population in a fish farm; an analysis of whether, based on model equation (2), we may expect a stable fish population from which harvesting could take place; and, if so, an analysis of how large an initial population of fish will be required to obtain this stable population and the length of time required for the stable population to be established (numerical solver recommended).

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