Question

Verify that the function y = a cos x + b sin x, where, a, b ∈ R is a solution of the differential equation d2y/dx2 + y=0...
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Answer 1

Step-by-step explanation:

ᵗʰᵉ ᵍⁱᵛᵉⁿ ᶠᵘⁿᶜᵗⁱᵒⁿ ⁱˢ ʸ = ᵃ ᶜᵒˢ ˣ + ᵇ ˢⁱⁿ ˣ … (1)

ᵈⁱᶠᶠᵉʳᵉⁿᵗⁱᵃᵗⁱⁿᵍ ᵇᵒᵗʰ ˢⁱᵈᵉˢ ᵒᶠ ᵉᵠᵘᵃᵗⁱᵒⁿ (1) ʷⁱᵗʰ ʳᵉˢᵖᵉᶜᵗ ᵗᵒ ˣ,

ᵈʸ/ᵈˣ = – ᵃ ˢⁱⁿˣ + ᵇ ᶜᵒˢ ˣ

ᵈ2ʸ/ᵈˣ2 = – ᵃ ᶜᵒˢ ˣ – ᵇ ˢⁱⁿˣ

ˡʰˢ = ᵈ2ʸ/ᵈˣ2 + ʸ

= – ᵃ ᶜᵒˢ ˣ – ᵇ ˢⁱⁿˣ + ᵃ ᶜᵒˢ ˣ + ᵇ ˢⁱⁿ ˣ

= 0

= ʳʰˢ

ʰᵉⁿᶜᵉ, ᵗʰᵉ ᵍⁱᵛᵉⁿ ᶠᵘⁿᶜᵗⁱᵒⁿ ⁱˢ ᵃ ˢᵒˡᵘᵗⁱᵒⁿ ᵒᶠ ᵗʰᵉ ᵍⁱᵛᵉⁿ ᵈⁱᶠᶠᵉʳᵉⁿᵗⁱᵃˡ ᵉᵠᵘᵃᵗⁱᵒⁿ.

ʰᵒᵖᵉ ⁱᵗ'ˢ ʰᵉˡᵖ ᵘʰ ❤️

Answer 2

Answer:

 constants a and b are:

 a = -8 / 65

 b = -1 / 65

Step-by-step explanation:

The first thing we must do in this case is find the derivatives:

 y = a sin (x) + b cos (x)

 y '= a cos (x) - b sin (x)

 y '' = -a sin (x) - b cos (x)

 Substituting the values:

 (-a sin (x) - b cos (x)) + (a cos (x) - b sin (x)) - 7 (a sin (x) + b cos (x)) = sin (x)

 We rewrite:

 (-a sin (x) - b cos (x)) + (a cos (x) - b sin (x)) - 7 (a sin (x) + b cos (x)) = sin (x)

 sin (x) * (- a-b-7a) + cos (x) * (- b + a-7b) = sin (x)

 sin (x) * (- b-8a) + cos (x) * (a-8b) = sin (x)

 From here we get the system:

 -b-8a = 1

 a-8b = 0

 Whose solution is:

 a = -8 / 65

 b = -1 / 65