First order differential eqn dy/dx + Py = Qy^n

In summary, the substitution z = y^-(n-1) transforms the general equation dy/dx + Py = Qy^n, where P and Q are functions of x, into the linear equation dz/dx - P(n-1)z = -Q(n-1). This is known as Bernoulli's equation and can be used to solve problems involving air flow. The correct procedure must be followed, including taking into account negative indices during differentiation.
  • #1
tony24810
42
0
show that the substitution z = y^-(n-1) transforms the general equation dy/dx + Py = Qy^n, where P and Q are functions of x, into the linear equation dz/dx - P(n-1)z = -Q(n-1). (Bernoulli's equation)


Well, I looked up Bernoulli's stuff on internet, found the usual air flow equation but not this one. In fact I followed the standard procedure to transform the equation, but what I got is a whole bunch of fraction rather than just the -(n-1).

Please help.
 
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  • #2
never mind

i managed to solved it myself, please ignore this post. i actually made mistake in my differentiation with the negative indices.
 

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FAQ: First order differential eqn dy/dx + Py = Qy^n

1. What is a first order differential equation?

A first order differential equation is an equation that relates the rate of change of a function to the function itself. It can be written in the form dy/dx = f(x,y), where dy/dx is the derivative of the function y with respect to x.

2. What does the term "P" represent in the equation dy/dx + Py = Qy^n?

The term "P" represents the coefficient of y in the differential equation. It is a constant that is typically dependent on the independent variable x.

3. What is the significance of the term "Q" in the equation dy/dx + Py = Qy^n?

The term "Q" represents the coefficient of y^n in the differential equation. It is also a constant that is typically dependent on the independent variable x.

4. How do you solve a first order differential equation of the form dy/dx + Py = Qy^n?

To solve this type of differential equation, you can use the method of separation of variables. This involves isolating the y and x terms on opposite sides of the equation and integrating both sides to find the general solution. You can then use initial conditions to find the particular solution.

5. Can you provide an example of a practical application of a first order differential equation?

One practical application of a first order differential equation is in modeling population growth. The equation can be used to describe the change in population over time, taking into account factors such as birth rate, death rate, and immigration. This information can then be used to make predictions and inform decision making in areas such as urban planning and resource management.

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