Check Differential Equation Solution

In summary, to verify the solution of the differential equation dX/dt=(2-x)(1-x), you can rewrite the derivative of the solution as 1/(2-x)=(2-x)(1-x) and solve for dX/dt. By plugging this into the original equation and simplifying, you can show that the solution is a true solution for the differential equation.
  • #1
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Homework Statement


Verify Differential Equation Solution
dX/dt=(2-x)(1-x)

Homework Equations


The solution is ln ((2-x)/(1-x))=t


The Attempt at a Solution


The derivative of the solution is -1/(2-x) dX/dt - (-1)/(1-x) dX/dt=1
But I plug this derivative in and I get stuck. How am I supposed to plug the solution and the derivative of the solution into the original differential equation? Can you help me set this up? I'm going to need a little help along the way...
 
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  • #2
-1/(2-x) dX/dt - (-1)/(1-x) dX/dt=1

rewrite it as

[tex]\left( \frac{-1}{2-x}+ \frac{1}{1-x} \right) \frac{dX}{dt}=1[/tex]

and bring the two terms in the bracket to the same common denominator.
 
  • #3
Thanks for the reply. I did that and my common denominator is (2-x).

With that, I solve for dX/dt and I get dX/dt=1/(2-x).

I plug the dX/dt into the original equation which looks like 1/(2-x)=(2-x)(1-x)

the (2-x) cancels from both sides and I am left with 1=1-x and then x=0.

I went wrong somewhere...
 
  • #4
I apologize...I found my mistake. The common denominator is
Code:
X[SUP]2[/SUP]-3x+2
which of course factors to (2-x)(1-x). I then multiply the numerator and denominator by the common denominator and put everything together. I then multiply both sides by (2-x)(1-x) to solve for dX/dt which leaves me with dX/dt=(2-x)(1-x). Plugging this into the original DE leaves me with (2-x)(1-x)=(2-x)(1-x) which proves that the solution is a true solution for this DE. Thanks for all of your help...
 

1. What is a differential equation solution?

A differential equation solution is a mathematical expression that satisfies a given differential equation. It represents the relationship between a function and its derivatives.

2. How do you check if a solution to a differential equation is correct?

To check if a solution to a differential equation is correct, you can substitute the solution into the original equation and see if it satisfies the equation. You can also take the derivative of the solution and see if it matches the derivative in the original equation.

3. What are the different methods for checking a differential equation solution?

There are various methods for checking a differential equation solution, including substitution, taking derivatives, using initial conditions, and graphing the solution to compare it with the original equation.

4. Can a differential equation solution be incorrect?

Yes, a differential equation solution can be incorrect if it does not satisfy the original equation or if it does not match the given initial conditions. It is important to check the solution to ensure its accuracy.

5. Why is it important to check the solution to a differential equation?

It is important to check the solution to a differential equation to ensure its accuracy and validity. A small error in the solution can lead to incorrect results and conclusions. Additionally, checking the solution can help identify any mistakes made during the solving process.

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