Differential Equation - Bernoulli Equation

In summary, the student is trying to solve a differential equation, but is stuck. They are following a wiki page to try to solve the equation, but are having difficulty. They are using partial fractions to solve for the various terms, but are getting stuck.
  • #1
pat666
709
0

Homework Statement


solve the differential equation
[tex] y'(t)=-4y+6y^3 [/tex]

Homework Equations





The Attempt at a Solution


I'm pretty sure (not positive) that this is a Bernoulli Equation.
I've been following this wiki in an attempt to solve: http://en.wikipedia.org/wiki/Bernoulli_differential_equation
[tex] (y'(t)=-4y+6y^3)/y^3 [/tex]
[tex] y'(t)/y^3=-4/y^2+6[/tex]
[tex] w=1/y^2 [/tex]
[tex] y'(t)*w/y=-4w+6[/tex]

I'm stuck here, also there's no x's or t's here so I am not sure that it is a Bernoulli Equation?

thanks for any help
 
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  • #2
dy/dt = 6y^3 - 4y. This is a variable separable differential equation, no Bernoulli pangs required. dy/(y)(6y^2 - 4) = dt. Integrate the LHS preferably using partial fractions and voila.
 
  • #3
[tex] dy/dt=-4y+6y^3 [/tex]
[tex]dy/(y(-4+6y^2))=dt[/tex]
[tex] u=y^2, du =2y dy [/tex]
[tex]=1/2 \int 1/(u(6u-4)) [/tex]
etc etc etc
[tex] t=1/8 (ln(3y^2-2)-2ln(y))+c [/tex]
[tex] 8t-c=ln((3y^2-2)/(y^2)) [/tex]
[tex] ce^t=(3y^2-2)/y^2 [/tex]

I am again stuck, how do I get y=----- out of this??

Thanks
 
  • #4
Mate, you need to use partial fractions to separate the denominator and integrate on each term separately.

gif.latex?\frac{1}{y(6y^{2}-4)}%20=%20\frac{A}{y}%20+%20\frac{B}{6y^{2}-4}.gif


Find A and B using the concept of partial fractions (this page is pretty neat...teaches you everything you need to know about them : http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/partialfracdirectory/PartialFrac.html), then multiply dy on both sides of the equation and integrate term by term.

gif.latex?\int\frac{dy}{y(6y^{2}-4)}%20=%20A\int\frac{dy}{y}%20+%20B\int\frac{dy}{6y^{2}-4}.gif
 
  • #5
pat666 said:
[tex] dy/dt=-4y+6y^3 [/tex]
[tex]dy/(y(-4+6y^2))=dt[/tex]
[tex] u=y^2, du =2y dy [/tex]
[tex]=1/2 \int 1/(u(6u-4)) [/tex]
etc etc etc
[tex] t=1/8 (ln(3y^2-2)-2ln(y))+c [/tex]
[tex] 8t-c=ln((3y^2-2)/(y^2)) [/tex]
[tex] ce^t=(3y^2-2)/y^2 [/tex]

I am again stuck, how do I get y=----- out of this??

Thanks

From
[tex]ce^t= \frac{3y^2- 2}{y^2}[/tex]
multiply both sides by [itex]y^2[/itex]:
[tex]ce^ty^2= 3y^2- 2[/tex]

[tex]3y^2- ce^ty^2= 2[/tex]
[tex](3- ce^t)y^2= 2[/tex]

Can you finish that?

(Does the problem require that you solve for x? Solutions to first order equations are often left as implicit functions.)
 
  • #6
certainly can:
[tex] y=sqrt(2/((3-ce^t))) [/tex]
I think that's right? and it keeps me away from partial fractions which I forgot about 3yrs ago.

Thanks svxx and Hallsofivy

edit:

oops, I used partial fractions in the integrations, guess I remember procedures and not names of said procedures...
 
Last edited:

1. What is the Bernoulli equation?

The Bernoulli equation is a type of differential equation that relates the rate of change of a dependent variable to its independent variables. It is commonly used in physics and engineering to model situations involving fluids, such as airflow or water flow.

2. What is the general form of the Bernoulli equation?

The general form of the Bernoulli equation is dy/dx + P(x)y = Q(x)yn, where y is the dependent variable, x is the independent variable, P(x) and Q(x) are functions of x, and n is a constant. This form is known as the Bernoulli differential equation.

3. How is the Bernoulli equation solved?

The Bernoulli equation can be solved by using a substitution technique, where a new variable z is defined as z = y1-n. This transforms the Bernoulli equation into a linear differential equation, which can be solved using standard methods such as separation of variables or integrating factors.

4. What are the applications of the Bernoulli equation?

The Bernoulli equation has many applications in physics and engineering, including fluid mechanics, aerodynamics, and thermodynamics. It is used to model air flow over airplane wings, water flow in pipes, and the movement of particles in a fluid.

5. What is the difference between the Bernoulli equation and the Euler equation?

The Bernoulli equation is a special case of the Euler equation, where the constant n is equal to 1. The Euler equation is a more general form of the Bernoulli equation, and it is used to model a wider range of phenomena, including the motion of fluids and the dynamics of vibrating strings.

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