anonymous
  • anonymous
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).
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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TuringTest
  • TuringTest
you want an exact solution without initial conditions?
TuringTest
  • TuringTest
or perhaps they just want the particular solution, which you can get with undetermined coefficients
anonymous
  • anonymous
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

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TuringTest
  • TuringTest
ok, so first you need to apply the method of undetermined coefficients are you at all familiar with that?
anonymous
  • anonymous
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?
TuringTest
  • TuringTest
no, I'll give you a decent link http://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx
TuringTest
  • TuringTest
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
anonymous
  • anonymous
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.
TuringTest
  • TuringTest
\[y_p'=-A\sin x+B\cos x\]\[y_p''=-A\cos x-B\sin x\]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
anonymous
  • anonymous
oh ok, that makes sense .. more sense atleast haha
TuringTest
  • TuringTest
\[-4A-3B=2\]\[2A-4B=0\]now look, it just became linear algebra
TuringTest
  • TuringTest
so solve for A and B and you will know a linear combination that will solve our problem
TuringTest
  • TuringTest
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
anonymous
  • anonymous
yes you did! thank you soo much!!!

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