Question

Let T : C∞(R) −→ C∞(R) be the map given by
T(f) = f′′− 2f′− 3f .

Find a linear combination a sin(x) + b cos(x) which solves the equation
f′′− 2f′− 3f = 2 cos(x).

Answer 1

you want an exact solution without initial conditions?

Answer 2

or perhaps they just want the particular solution, which you can get with undetermined coefficients

Answer 3

i think so .. this part of the question just asks for a linear combo of sinx and cosx which solves the equation .. then it asks to check if e^-x and e^3x are in the ker, then it says find a sol'n to the same questiona nd gives f(0)=2 and fprime (0) = 3

Answer 4

ok, so first you need to apply the method of undetermined coefficients
are you at all familiar with that?

Answer 5

no, my prof gave us this homework but we haven't even looked at it yet.
is that a method in which i'd be able to find if i googled it hah?

Answer 6

no, I'll give you a decent link
http://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx

Answer 7

basically you are going to guess that there is some solution in the form\[y_p=A\cos x+B\sin x\]and plug that into the equation

Answer 8

the only thing is this is for a lin. alg. course, so i'm not sure if i'm supposed to be using calculus to solve it.

Answer 9

\[y_p'=-A\sin x+B\cos x\]\[y_p''=-A\cos x-B\sin x\]plug that into the original\[y''− 2y'− 3y\]\[ =-A\cos x-B\sin x+2A\sin x-2B\cos x-3A\cos x-3B\sin x\]\[=(-4A-3B)\cos x+(2A-4B)\sin x=2\cos x\]well we are about to get a system from this so hang on
let's compare coefficients of sin and cos on each side

Answer 10

oh ok, that makes sense .. more sense atleast haha

Answer 11

\[-4A-3B=2\]\[2A-4B=0\]now look, it just became linear algebra

Answer 12

so solve for A and B and you will know a linear combination that will solve our problem

Answer 13

I have to go, I hope that helped a bit
the other part, just use the characteristic eqn I guess and you will see that C1e^-x and C2e^3x are solutions
then use the initial conditions about y(0) and y'(0) to find C1 and C2, which will again be linear algebra-ish

Answer 14

yes you did! thank you soo much!!!