Solving for 'k' such that some function is a solution of a diff. eq.

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In summary, to find the value of k such that x(t)=18^t is a solution of the differential equation dx/dt=kx, you need to set k equal to ln(18).
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mesa

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Homework Statement



Find k such that x(t)=18^t is a solution of the differential equation dx/dt=kx

The Attempt at a Solution



I took the derivative of x(t)

x'= ln(18)*18^t

then set it equal to kx,

kx = ln(18)*18^t

giving,

k = [ln(18)*18^t]/x

I am sure I am missing something simple but have not been able to figure it out.
 
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  • #2
mesa said:

Homework Statement



Find k such that x(t)=18^t is a solution of the differential equation dx/dt=kx

The Attempt at a Solution



I took the derivative of x(t)

x'= ln(18)*18^t

then set it equal to kx,

kx = ln(18)*18^t

giving,

k = [ln(18)*18^t]/x

I am sure I am missing something simple but have not been able to figure it out.

Your problem is here :

##kx = ln(18)18^t##

You have ##x(t) = 18^t## so really you have :

##k(18^t) = ln(18)18^t##
##k = ln(18)##
 
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  • #3
Zondrina said:
Your problem is here :

##kx = ln(18)18^t##

You have ##x(t) = 18^t## so really you have :

##k(18^t) = ln(18)18^t##
##k = ln(18)##

That's embarrassing... :)

Thank you!
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes how a function changes over time, based on the rate of change of the function. It involves both the function itself and one or more of its derivatives.

2. Why do we need to solve for 'k' in a differential equation?

In a differential equation, 'k' represents a constant that is necessary to find the exact solution of the equation. It is often used to represent initial or boundary conditions that are required to fully define the problem.

3. How do we solve for 'k' in a differential equation?

The process of solving for 'k' in a differential equation involves using algebraic manipulation and integration techniques to isolate the constant 'k' on one side of the equation. This may require multiple steps and the application of various mathematical rules.

4. What are the different methods for solving a differential equation for 'k'?

There are several methods for solving a differential equation for 'k', including the method of undetermined coefficients, variation of parameters, and the method of reduction of order. The appropriate method to use depends on the specific form of the equation and the information given.

5. What are some practical applications of solving for 'k' in a differential equation?

Solving for 'k' in a differential equation is a fundamental tool in many fields of science, including physics, engineering, and biology. It can be used to model and predict the behavior of systems over time, such as the growth of a population, the spread of a disease, or the movement of objects under the influence of forces.

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