Question

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}

Answer 1

Answer:

  y =C₁eˣ + C₂e⁻ˣ  + C₃ cos x + C₄ sin x

Step-by-step explanation:

given,

y⁽⁴⁾ - y = 0

( D⁴ - 1 ) y = 0..............(1)

writing characteristics equation

m⁴ -1  =  0

(m² - 1)(m² +1) = 0

(m - 1)(m+1)(m²+1) = 0

hence, roots of the equation comes out to be

              = 1, -1, ± i

so the general solution will be

  y =C₁eˣ + C₂e⁻ˣ  + C₃ cos x + C₄ sin x