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

refer the attachment :)