Differential Equations - Verifying a solution of a given DE

In summary, the given problem involves verifying if a given function is a solution to a given differential equation. The function \ln{\frac{2-x}{1-x}}=t is to be checked for this purpose. To solve this, the derivative of the given function is taken, which simplifies to (2-x)(1-x). This confirms that the given function is indeed a solution to the given differential equation.
  • #1
Nickg140143
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Homework Statement



Verify that the indicated funciton is a solution of the given Differential Equation. c1 and c2 denote constants where appropriate.

[tex]\frac { dX }{ dt } =(2-x)(1-x);\quad \quad \ln { \frac { 2-x }{ 1-x } } =t[/tex]

The Attempt at a Solution


I'm not quite sure how to really start this problem. If I'm reading the question right, the differential equation is

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

and that the solution I'm checking is
[tex]\ln { \frac { 2-x }{ 1-x } } =t[/tex]

I was thinking of perhaps integrating
[tex]\frac { dX }{ dt } =(2-x)(1-x)[/tex]
then maybe plug in the the given value of t?

I've read through the section and looked at my notes from class, but I can't seem to fully understand what I should be doing in this problem. The solution in the back of the book says

[tex]\frac { d }{ dt } \ln { \frac { 2-X }{ 1-X } } =1,\quad \left[ \frac { -1 }{ 2-X } +\frac { 1 }{ 1-X } \right] \frac { dX }{ dt } =1[/tex]
Simplifies to
[tex]\frac { dX }{ dt } =(2-X)(1-X)[/tex]

...Not entirely sure what it necessarily means by that though. Any help would be GREATLY appreciated.
 
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  • #2
I think they just want you to take the derivative of [tex] \ln { \frac { 2-x }{ 1-x } } [/tex] and find that it does indeed equal [tex] (2-x)(1-x) [/tex]

EDIT: although when I take the derivative I get the answer upside down. I get [tex] \frac{1}{(1-x)(2-x)} [/tex]
 
Last edited:
  • #3
You just need to take the derivative of the natural log function. Some things you might want to keep in mind when doing this are that you can split this function into two natural logs, and also the chain rule.
 
  • #4
Alright, I'll give that a go and try to make sense of the question, thanks for the help you two
 

1. What is a differential equation?

A differential equation is a mathematical equation that involves the derivatives of a function. It is used to model relationships between variables that change continuously over time or space.

2. How do you verify a solution of a given differential equation?

To verify a solution of a differential equation, you substitute the solution into the original equation and check if the equation holds true. If it does, then the solution is valid.

3. What is the order of a differential equation?

The order of a differential equation is the highest derivative present in the equation. For example, a first-order differential equation contains only first derivatives, while a second-order differential equation contains second derivatives.

4. What is the difference between an ordinary and a partial differential equation?

An ordinary differential equation involves a single independent variable, while a partial differential equation involves multiple independent variables. Ordinary differential equations are used to model relationships in one-dimensional systems, while partial differential equations are used for multi-dimensional systems.

5. Can a differential equation have multiple solutions?

Yes, a differential equation can have multiple solutions. In some cases, the general solution of a differential equation can be expressed in terms of a constant, which can take on different values and result in different solutions.

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