Calculus II Differential Equation Question

In summary, the conversation discusses finding constants a and b in a given differential equation so that both y=e^x and y=e^(2x) are solutions. The solution involves plugging in the given solutions and factoring out the exponentials to get two equations, which can then be solved for a and b. The final solution is a=1/2 and b=-3/2.
  • #1
member 508213

Homework Statement


We are just starting to learn about basic differential equations in Calc 2 and I learned about separable differential equations but I'm not familiar with this style, here's the question:

Given the differential equation of the form ay"+by'+y=0, find constants a and b so that both y=e^x and y=e^(2x) are solutions.

Homework Equations


None really

The Attempt at a Solution


To be honest I have thought about this a lot and I'm not exactly sure where to start. I can't see how you could use separable differentials to solve this one and I really am not too sure how to "solve" it. I was thinking about taking the integral of both sides but I'm not sure that would be a correct operation since there is no "dy". I'm also not exactly sure how you could solve that equation to get e^x in the first place so I'm pretty lost on this one.
 
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  • #2
Austin said:

Homework Statement


We are just starting to learn about basic differential equations in Calc 2 and I learned about separable differential equations but I'm not familiar with this style, here's the question:

Given the differential equation of the form ay"+by'+y=0, find constants a and b so that both y=e^x and y=e^(2x) are solutions.

Homework Equations


None really

The Attempt at a Solution


To be honest I have thought about this a lot and I'm not exactly sure where to start. I can't see how you could use separable differentials to solve this one and I really am not too sure how to "solve" it.
You don't need to solve this DE since they have already given you the solutions. Just plug each solution into your given DE, and that will give you two equations in a and b.
Austin said:
I was thinking about taking the integral of both sides but I'm not sure that would be a correct operation since there is no "dy". I'm also not exactly sure how you could solve that equation to get e^x in the first place so I'm pretty lost on this one.
 
  • #3
So I plugged each solution into the DE and get:

ae^x + be^x + e^x=0 and 4ae^(2x) + 2be^(2x) + e^(2x)=0

I apologize if what I should do next to solve for a & b is obvious but I don't see what to do?? I feel like you do not know enough to solve for a & b?
 
  • #4
Factor the exponentials out of both equations.
 
  • #5
Oh I see. I did think of factoring out the exponentials but for whatever reason I did not think of removing them to get the two equations. I got that a=1/2 and b =-3/2

Thanks
 

FAQ: Calculus II Differential Equation Question

1. What is the purpose of studying Calculus II Differential Equations?

Calculus II Differential Equations is a branch of mathematics that deals with the relationships between functions and their rates of change. It is used to model and solve real-world problems in fields such as engineering, physics, and economics.

2. What are the main concepts covered in Calculus II Differential Equations?

The main concepts covered in Calculus II Differential Equations include solving differential equations, techniques for finding solutions, and applications of differential equations such as growth and decay, population dynamics, and harmonic motion.

3. What are the key differences between ordinary and partial differential equations?

Ordinary differential equations involve one independent variable and its derivatives, while partial differential equations involve multiple independent variables and their partial derivatives. In other words, ordinary differential equations deal with functions of one variable, while partial differential equations deal with functions of multiple variables.

4. What are some common methods for solving differential equations?

Some common methods for solving differential equations include separation of variables, integrating factors, substitution, and using power series. Each method is useful for different types of differential equations and can be applied depending on the specific problem at hand.

5. How can I apply Calculus II Differential Equations in real life?

Calculus II Differential Equations can be applied in various fields such as physics, engineering, economics, and biology. For example, it can be used to model the growth of populations, the motion of objects under the influence of forces, and the changing values of financial investments over time.

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