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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6
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 ❤️
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19
Answer:
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.
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