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

y =C₁eˣ + C₂e⁻ˣ + C₃ cos x + C₄ sin x

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