Linear First Order Differential Equation - Mixture Problem

In summary, the solution to a linear differential equation consists of the sum of the general solution of the homogeneous part and an arbitrary particular solution of the inhomogeneous equation. The particular solution can be found by setting the derivative equal to zero and solving for the constant. The homogeneous solution is of the form P=a eλt, where λ can be found by solving for the constant.
  • #1
jdinatale
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0
The problem and attempt at solution are typed below

paint-1.jpg
 
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  • #2
The solution of a linear differential equation as yours is the sum of the general solution of the homogeneous part (dP/dt+0.72P=0) and an arbitrary particular solution of the inhomogeneous equation. That solution can be for which dP/dt=0, that is P=const. Find that constant.
The solution of the homogeneous equation is of the form P=a eλt. Find λ.

ehild
 
  • #3
ehild said:
The solution of a linear differential equation as yours is the sum of the general solution of the homogeneous part (dP/dt+0.72P=0) and an arbitrary particular solution of the inhomogeneous equation. That solution can be for which dP/dt=0, that is P=const. Find that constant.
The solution of the homogeneous equation is of the form P=a eλt. Find λ.

ehild

Thanks ehild, but I think what would help me is if someone could point out the flaws in my original solution. I'm trying to solve this problem using the method taught in class so I'm hesitant to try something that we haven't learned.

I don't see any mistakes in my work.
 

Related to Linear First Order Differential Equation - Mixture Problem

What is a linear first order differential equation?

A linear first order differential equation is a mathematical equation that relates a function to its first derivative. It has the general form y' + P(x)y = Q(x), where P(x) and Q(x) are functions of x.

What is a mixture problem in terms of differential equations?

A mixture problem is a type of problem in which we are given two or more substances and we need to find the amount or concentration of each substance in the final mixture. This type of problem can be solved using differential equations, specifically linear first order differential equations.

How do you solve a linear first order differential equation - mixture problem?

To solve a linear first order differential equation - mixture problem, we need to follow these steps:

  1. Identify the substances and their initial amounts or concentrations.
  2. Write a differential equation that relates the rate of change of the total amount or concentration of the mixture to the individual rates of change of each substance.
  3. Solve the differential equation using separation of variables or an integrating factor.
  4. Use the obtained solution to find the values of the substances at a specific time or concentration.

What are some real-life applications of linear first order differential equations - mixture problems?

Linear first order differential equations - mixture problems have many real-life applications, such as:

  • Chemistry: determining the concentration of substances in a chemical reaction
  • Pharmacology: calculating the concentration of drugs in the body over time
  • Environmental science: studying the mixing of pollutants in air or water
  • Economics: analyzing the flow of money in a market with multiple currencies

Can we use linear first order differential equations to solve mixture problems with more than two substances?

Yes, linear first order differential equations can be used to solve mixture problems with any number of substances. The process is the same as for two substances, but the differential equation will have more terms for each additional substance. The solution can also be obtained using matrix methods.

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