Differential Equation Root of Pt (aka xy)

In summary, the conversation discusses the process of finding the integral of dP/dt = sqrt(Pt) with the initial condition P(1) = 2. The solution involves separating the equation and using the method of integration to find the solution.
  • #1
avant_t7
3
0

Homework Statement


dP/dt=root of (Pt)


Homework Equations


P(1)=2


The Attempt at a Solution


Well, I figure that the integrating factor is 1, since there is no P(x) value, so it's a matter of finding the integral of root(Pt). I can't solve this because the only method I know is substitution, and it doesn't work on this equation.

Thanks guys
 
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  • #2
avant_t7 said:

Homework Statement


dP/dt=root of (Pt)


Homework Equations


P(1)=2


The Attempt at a Solution


Well, I figure that the integrating factor is 1, since there is no P(x) value, so it's a matter of finding the integral of root(Pt). I can't solve this because the only method I know is substitution, and it doesn't work on this equation.

Thanks guys
this equation is separable. Write sqrt(Pt) as P1/2t1/2.
 
  • #3
I see that, I guess I'm just confused by finding the integral of two variables. If I substitute u for Pt, and du=(t)dP/dt + P dt...

will that work?
 
  • #4
You don't need to do a substitution and you aren't going to have any integrals with two variables.

[tex]\frac{dP}{dt} = \sqrt{Pt} = P^{1/2}t^{1/2}[/tex]

Separate to get the following.
[tex]\frac{dP}{P^{1/2}} = t^{1/2} dt[/tex]

Now integrate.
 
  • #5
shesh you're right, how silly of me.

THanks a lot Mark!
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives (i.e. the rate of change of the function). It is commonly used to model systems in which the rate of change of a variable is dependent on the value of that variable itself.

2. What is the root of a differential equation?

The root of a differential equation is any value of the independent variable (usually denoted as x) that satisfies the equation. In other words, it is the value of x that makes the equation true when substituted into the equation.

3. How are differential equations used in science?

Differential equations are used in many fields of science, including physics, chemistry, biology, and engineering. They are used to model various physical and biological processes, such as population growth, chemical reactions, and electrical circuits.

4. What is Pt (aka xy) in a differential equation?

Pt (aka xy) represents the dependent variable in a differential equation. It is the variable whose value is determined by the independent variable (x) and its derivatives.

5. How do you find the root of a differential equation?

Finding the root of a differential equation involves solving the equation for the value of the independent variable (x) that makes the equation true. This can be done analytically using mathematical methods or numerically using computer algorithms.

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