Understanding and Solving DiffEq: x' = 2*x^(1/2)

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In summary, the given differential equation is separable and can be written as dx/dt = 2x1/2. By integrating both sides and including a constant of integration, the solution x = t2 can be obtained, as well as the solution provided by Wolfram, 1/4 (4 t2 + 4 t C + C2).
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pergradus
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Homework Statement



I have the differential equation:

x' = 2*x1/2

The Attempt at a Solution



I can see easily that the solution x = t2 satisfies the equation, however wolfram tells me the solution is:

1/4 (4 t2 + 4 t C + C2)

which also satisfies the solution... I'm wondering how do I solve this differential equation in order to get the solution Mathematic got?
 
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  • #2
pergradus said:

Homework Statement



I have the differential equation:

x' = 2*x1/2

The Attempt at a Solution



I can see easily that the solution x = t2 satisfies the equation, however wolfram tells me the solution is:

1/4 (4 t2 + 4 t C + C2)

which also satisfies the solution... I'm wondering how do I solve this differential equation in order to get the solution Mathematic got?
This DE is separable. Write it as
dx/dt = 2x1/2
=> dx/(2x1/2) = dt

Now integrate both sides. Don't forget the constant of integration.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates the rate of change of a variable to its current value. In other words, it describes how a variable changes over time.

2. What does x' = 2*x^(1/2) mean?

This is a specific type of differential equation called a separable differential equation. It means that the rate of change of a variable x is equal to two times the square root of x. In order to solve this equation, we need to separate the variables on either side of the equation and then integrate.

3. How do you solve this differential equation?

To solve this differential equation, we need to separate the variables and then integrate both sides. This will give us an expression for x in terms of t. We can then solve for any initial conditions and use the solution to make predictions about the behavior of x over time.

4. What is the significance of x^(1/2) in this equation?

The term x^(1/2) represents the square root of x. It is significant because it describes the rate of change of x, which is directly proportional to the square root of x. This means that as x increases, the rate of change also increases, but at a decreasing rate.

5. Can this differential equation be applied to real-world situations?

Yes, this differential equation can be applied to real-world situations where the rate of change of a variable is proportional to its current value. For example, it can be used to model population growth, radioactive decay, or the spread of disease.

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