Linear and Bernoulli First Order

In summary, the conversation is about solving three different types of differential equations: linear, Bernoulli's, and a combination of both. The problem statement includes three given equations and the relevant equations for solving LDE and BDE. The person encountered some difficulties in solving the equations, but eventually found the correct solutions for each one.
  • #1
darkmagic
164
0
1. The problem statement.

1. http://img194.imageshack.us/img194/3815/linear.png

2. http://img25.imageshack.us/img25/1543/bernoulli1.jpg

3. http://img23.imageshack.us/img23/7991/bernoulli2.jpg

2.Relevant equations

Problem 1 is Linear Differential Equation and Problems 2 and 3 are Bernoulli's D.E.

LDE is given by: dy/dx + yP(x) = Q(x)
BDE is given by: dy/dx + yP(x) = Q(x)y^n

The Attempt at a Solution



On those three equations, I'm trying to make the equations as exactly to the format on the LDE and BDE. However, I cannot get P(x) and Q(x) on no.1. The y is always y^3. In no.2, there will be Q(x)y^n and there is no y. In no.3, I solved that. However, that problem is BDE and I solved it using LDE with dx/dy. Do you think there is something wrong with the equation?

Thank you.
 
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  • #2
I think I solved the three equations. There's something wrong with those equations. The first problem can be solved by using Bernoulli but it is under the linear problem set. I got the same answer exactly as the provided answer. The second problem is Bernoulli. I modified the equation and when I solved it, I got the same answer exactly as the provided answer. The equation goes 2x^3y'=y(y^2+3x^2). The third problem is the same as the first. Instead of Bernoulli, it is solvable using Linear. Anyway, thank you.
 

FAQ: Linear and Bernoulli First Order

What is the difference between linear and Bernoulli first order differential equations?

Linear first order differential equations can be written in the form y' + p(x)y = g(x), where p(x) and g(x) are functions of x. Bernoulli first order differential equations can be written in the form y' + p(x)y = g(x)y^n, where n is a constant. This means that the dependent variable y is raised to a power in Bernoulli equations, making them nonlinear.

How are linear and Bernoulli first order differential equations solved?

Linear first order differential equations can be solved using the integrating factor method or by separation of variables. Bernoulli first order differential equations can be solved by using the substitution u = y^(1-n) to transform them into linear equations.

What is the significance of the integrating factor in solving linear first order differential equations?

The integrating factor is a function that is multiplied to both sides of the equation to make it easier to solve. It is typically found by multiplying both sides of the equation by e^(∫p(x)dx), where p(x) is the coefficient of y. This results in one side of the equation being a derivative of e^∫p(x)dx, which can then be solved by integration.

Can linear and Bernoulli first order differential equations have initial conditions?

Yes, both linear and Bernoulli first order differential equations can have initial conditions, which are values of the dependent variable at a specific point. These initial conditions can be used to find the particular solution to the differential equation.

How are linear and Bernoulli first order differential equations used in real-world applications?

Linear and Bernoulli first order differential equations are used to model and solve various real-world problems, such as population growth, radioactive decay, and chemical reactions. These equations can also be used in physics, engineering, and economics to describe the relationship between different variables and their rates of change.

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