Help me find a function

[Tenshi]

How can I find the function f(x) if i know that:

f"(x) - f(x)=2e^x , f(0)=-1 f'(0)=2

 

[Deleted member 4993]

How can I find the function f(x) if i know that:
f"(x) - f(x)=2e^x , f(0)=-1 f'(0)=2

This is a second order ordinary differential equation (ODE) with homogeneous solution and particular solution.

What is the homogeneous solution of the given ODE?

What is the particular solution of the given ODE?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

 

[HallsofIvy]

I'll give you a further hint. The "associated homogeneous equation" is f''- f= 0. If you try a solution of the form f(x)= e^(ax) then f'= ae^(ax) and f''= a^2e^(ax) so the equation becomes f''- f= a^2e^(ax)- e^(ax)= e^(ax)(a^2- 1)= 0. e^(ax) is never 0 so what must a equal?

So what is the general solution to the associated homogeneous equation?

What do you get if you also try f(x)= Axe^x where A is a constant?

Now, I suspect that, if you have been given a problem like this, you must be taking an introductory course in "Differential Equations".
So why don't you already know all this?