Simple differential equation question

In summary, a simple differential equation is an equation that involves an unknown function and its derivatives. The purpose of solving it is to find the unknown function that satisfies the equation and can be done by finding the general solution using methods such as separation of variables or substitution. These equations have many real-life applications and can have multiple solutions depending on the type of equation and initial conditions.
  • #1
thursday
2
0
is there anyone who can help me.

how do I solve the following differential equation:

dP = P^2V
dV

im not sure what the intergral of 1/P^2 is


is the answer P = Ce^(v^2) - Ce to the power of v squared ?
thanx
 
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  • #2
1/P2= P-2. Do you know the integral of that?

"is the answer P = Ce^(v^2) - Ce to the power of v squared ?"

No, it's not anything like that. For one thing, it does not involve any exponentials.
 
  • #3


Sure, I can help you with this differential equation question. To solve it, we can use the method of separation of variables. First, let's rewrite the equation as:

dP/dV = P^2V

Next, we can separate the variables by moving all the terms with P to one side and all the terms with V to the other side:

dP/P^2 = VdV

Now, we can integrate both sides:

∫dP/P^2 = ∫VdV

To find the integral of 1/P^2, we can use the power rule for integration, which states that ∫x^n dx = x^(n+1)/(n+1) + C. In this case, n = -2, so the integral of 1/P^2 becomes P^-1 + C.

Substituting this into our equation, we get:

P^-1 + C = V^2/2 + C

Next, we can solve for P by isolating it on one side:

P^-1 = V^2/2 + C - C

P^-1 = V^2/2

Finally, we can take the reciprocal of both sides to get the solution for P:

P = 2/V^2

So the solution to the differential equation is P = 2/V^2 + C. I'm not sure where you got the answer P = Ce^(v^2) - Ce^(v^2) from, but it does not appear to be the correct solution. I hope this helps!
 

Related to Simple differential equation question

1. What is a simple differential equation?

A simple differential equation is an equation that involves an unknown function and its derivatives. It can be written in the form of f'(x) = g(x), where f'(x) represents the derivative of the function f(x) and g(x) represents the function itself.

2. What is the purpose of solving a simple differential equation?

The purpose of solving a simple differential equation is to find the unknown function that satisfies the given equation. This can help in understanding and predicting the behavior of various systems in fields such as physics, engineering, and economics.

3. How do you solve a simple differential equation?

Solving a simple differential equation involves finding the general solution, which is the most general form of the equation that satisfies all possible values of the constants involved. This can be done using various methods, such as separation of variables, substitution, or using an integrating factor.

4. Are there any real-life applications of simple differential equations?

Yes, there are many real-life applications of simple differential equations. They are used to model and understand phenomena in various fields, such as population growth, chemical reactions, and electrical circuits.

5. Can a simple differential equation have multiple solutions?

Yes, a simple differential equation can have multiple solutions. This is because the general solution contains constants that can take on different values, resulting in different specific solutions. However, the number of solutions is dependent on the type of differential equation and the initial conditions given.

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