Differential Equations - Bernoulli equation

In summary, the conversation discusses a differential equation problem involving the logistic equation. The problem asks to find the solution for two given initial values, the time when one of the solutions reaches a specific value, and when the other solution becomes infinite. One approach is to use the Bernoulli differential equation method, while another involves solving for y using logarithms.
  • #1
ajkess1994
9
0
Afternoon, anyone that would like to take a look at this Differential Equation problem it would be very helpful. I have tried separating the problem, but I am only working with one known term.

Consider the logistic equation

$$\dot{y}=y(1-y). $$

(a) Find the solution satisfying $y_1(0)=6$ and $y_2(0)=−1. $

$y_1(t)= ?$

$y_2(t)= ?$

(b) Find the time $t$ when $y_1(t)=3. $

$t= ?$

(c) When does $y_2(t)$ become infinite?

$t= ?$
 
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  • #2
ajkess1994 said:
Afternoon, anyone that would like to take a look at this Differential Equation problem it would be very helpful. I have tried separating the problem, but I am only working with one known term.

Consider the logistic equation

[y˙=y(1−y)] NOTICE: There is a small dot above the first y term, NOT off to the side

(a) Find the solution satisfying y1(0)=6 and y2(0)=−1.

y1(t)= ?

y2(t)= ?

(b) Find the time t when [y1(t)=3].

t= ?

(c) When does y2(t) become infinite?

t= ?
To get you started: This is a Bernoulli differential equation. You can see the solution method here.

-Dan
 
  • #3
Or another approach:
\begin{align*}
\dot{y}&=y(1-y) \\
\int\frac{dy}{y(1-y)}&=\int dt \\
\int\frac{dy}{y}-\int\frac{dy}{y-1}&=t+C \\
\ln|y|-\ln|y-1|&=t+C \\
\ln\left|\frac{y}{y-1}\right|&=t+C \\
\frac{y}{y-1}&=Ce^t \\
&\vdots
\end{align*}
You can solve for $y$ from here.
 
  • #4
Thank you Dan

- - - Updated - - -

Thank you Ackbach
 

Related to Differential Equations - Bernoulli equation

1. What is the Bernoulli equation and what does it represent?

The Bernoulli equation is a first-order ordinary differential equation that relates the changes in pressure, velocity, and height of a fluid as it moves along a streamline. It represents the conservation of energy in a fluid flow system.

2. How is the Bernoulli equation derived?

The Bernoulli equation is derived from the principle of conservation of energy, where the sum of kinetic, potential, and flow energies remains constant along a streamline in an ideal fluid flow. It is also derived using the Navier-Stokes equations and the continuity equation.

3. What are the applications of the Bernoulli equation?

The Bernoulli equation has various applications in fluid mechanics, such as in calculating airspeed and lift in aerodynamics, determining flow rates in pipes, and analyzing the flow of blood in arteries. It is also used in the design of aircraft wings, turbines, and other devices that involve fluid flow.

4. What are the assumptions made in the Bernoulli equation?

The Bernoulli equation assumes that the fluid is incompressible, inviscid, and steady-state. It also assumes that the flow is along a streamline, and there is no heat transfer or external work being done on the fluid.

5. What are the limitations of the Bernoulli equation?

The Bernoulli equation is only applicable to ideal fluid flow, which is a simplified model that does not account for factors such as viscosity, turbulence, and compressibility. It also cannot be used for flows with significant changes in elevation or when there are external forces acting on the fluid.

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