Calculating Differential Equations for Exponential Functions

In summary, given the rate of change of the amount of material in a jar is proportional to the square of the amount present with proportionality constant k=-3, a differential equation is formed (dy/dt=-3y^2). By separating variables, integrating both sides, and solving for y, the solution to the differential equation is y=1/(3t+C*). Given y=1 when t=1/3, the value of the constant C* can be determined as -1. Therefore, the final solution is y=1/(3t-1).
  • #1
joev714
2
0

Homework Statement



The rate of change of the amount of material in a jar is proportional to the square of the amount present with proportionality constant k=-3.

i) Write a differential equation for this situation
ii) Solve the differential equation AND find y if y=1 when t=1/3

Homework Equations


Not sure of the specific equation, but I know it has something to do with exponential equations


The Attempt at a Solution



dy/dt=-3y^2

seperate variables, integrate both sides, That's as far as I got, sorry :[
 
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  • #2
joev714 said:

Homework Statement



The rate of change of the amount of material in a jar is proportional to the square of the amount present with proportionality constant k=-3.

i) Write a differential equation for this situation
ii) Solve the differential equation AND find y if y=1 when t=1/3

Homework Equations


Not sure of the specific equation, but I know it has something to do with exponential equations


The Attempt at a Solution



dy/dt=-3y^2

seperate variables, integrate both sides, That's as far as I got, sorry :[
Your diff. equation is right. When you separated variables, what did you get?
 
  • #3
dy/y^2=-3dt

Integrating both sides yielded -1/y=-3t+C
 
  • #4
joev714 said:
dy/y^2=-3dt

Integrating both sides yielded -1/y=-3t+C
So 1/y = 3t - C = 3t + C*, where C* is just another constant.
Now solve for y.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates the rate of change of a function to the function itself. It involves derivatives, which represent the instantaneous rate of change of a function at a given point.

2. What is the purpose of studying differential equations in Calc AB?

In Calc AB, differential equations are used to model real-world problems and analyze the behavior of systems. They are also an important tool for solving optimization and related rates problems.

3. How are differential equations solved in Calc AB?

Differential equations can be solved using various methods, including separation of variables, integrating factors, and substitution. Calculus concepts such as derivatives and integrals are used extensively in the solving process.

4. Can differential equations be used in other branches of science?

Yes, differential equations are widely used in various fields of science, including physics, chemistry, biology, and engineering. They are used to describe and analyze physical systems and phenomena.

5. Are there any applications of differential equations in everyday life?

Yes, there are many real-life applications of differential equations, such as predicting population growth, modeling the spread of diseases, and analyzing the stock market. They are also used in fields like economics, medicine, and meteorology.

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