How Can You Eliminate Constants in a Differential Equation?

[th3chemist]

Homework Statement

eliminating c1 and c2 from y, y' and y'' to create a differential equation

Homework Equations

I have the equation y = (c1 +c2x)e^x
y' = c2e^x +(c1+c2x)e^x
y'' = c2e^x + c2e^x + (c1+c2x)e^x)

(from product rule)

The Attempt at a Solution

I'm just not sure how to eliminate the constants. If I try to do y'' - y i still have constants left.

 

[SammyS]

Homework Statement

eliminating c1 and c2 from y, y' and y'' to create a differential equation

Homework Equations

I have the equation y = (c1 +c2x)e^x
y' = c2e^x +(c1+c2x)e^x
y'' = c2e^x + c2e^x + (c1+c2x)e^x)

(from product rule)

The Attempt at a Solution

I'm just not sure how to eliminate the constants. If I try to do y'' - y i still have constants left.

What do you get if you plug y = (c₁ +c₂x)e^x for the second term in y' ?

Solve that for c₂.

 

[bigfooted]

with y"-y you are only using part of you available information.
What happens if you subtract y-y' = ... ?
or y'-y" ?

 

[th3chemist]

What do you get if you plug y = (c₁ +c₂x)e^x for the second term in y' ?

Solve that for c₂.

thank you! I can't believe I didn't see that.

I got c2 = (y' - y)/ e^x. I can then sub this into y'' to get the differential.

Thank you :)

 

[lendav_rott]

thank you! I can't believe I didn't see that.

I got c2 = (y' - y) e^x. I can then sub this into y'' to get the differential.

Thank you :)

Isn't it c2 = (y' - y) e^-x ?

 

[th3chemist]

Isn't it c2 = (y' - y) e^-x ?

yes yes, for some reason it didn't put the / button