Calculus II Differential Equation Question

[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.

 

[Mark44]

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.

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.

 

[member 508213]

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?

 

[LCKurtz]

Factor the exponentials out of both equations.

 

[member 508213]

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