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Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. (please explain your work)

y(4) - y =0 , {e^x,e^-x,cosx,sinx}

Respuesta :

Answer:

Ā  y =Cā‚eĖ£ + Cā‚‚eā»Ė£ Ā + Cā‚ƒ cos x + Cā‚„ sin x

Step-by-step explanation:

given,

yā½ā“ā¾ - y = 0

( Dā“ - 1 ) y = 0..............(1)

writing characteristics equation

mā“ -1 Ā = Ā 0

(mĀ² - 1)(mĀ² +1) = 0

(m - 1)(m+1)(mĀ²+1) = 0

hence, roots of the equation comes out to be

Ā  Ā  Ā  Ā  Ā  Ā  Ā  = 1, -1, Ā± i

so the general solution will be

Ā  y =Cā‚eĖ£ + Cā‚‚eā»Ė£ Ā + Cā‚ƒ cos x + Cā‚„ sin x

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