Examples

Chapter 9 Class 12 Differential Equations
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Example 3 Verify that the function π¦=π cosβ‘γπ₯+π sinβ‘γπ₯, γ γ where , π, πβπ is a solution of the differential equation (π^2 π¦)/(ππ₯^2 )+π¦=0 π¦=π cosβ‘γπ₯+π sinβ‘γπ₯ γ γ ππ¦/ππ₯=π/ππ₯ (π cosβ‘γπ₯+π sinβ‘γπ₯ γ γ ) =π π(cosβ‘π₯ )/ππ₯+π π(sinβ‘π₯ )/ππ₯ =π(γβsinγβ‘π₯ )+π(cosβ‘π₯ ) =βπ π πππ₯+π πππ π₯ (π^2 π¦)/(ππ₯^2 )=π/ππ₯ (ππ¦/ππ₯) =π/ππ₯ (βπ π πππ₯+π πππ π₯) =βπ π(sinβ‘π₯ )/ππ₯+π (π(cosβ‘π₯))/ππ₯ =βπ(cosβ‘π₯ )+π(βsinβ‘π₯) =βπ cosβ‘γπ₯βπ sinβ‘π₯ γ Now, we have to verify (π^2 π¦)/(ππ₯^2 )+π¦=0 Taking L.H.S (π^2 π¦)/(ππ₯^2 )+π¦ =(βπ cosβ‘γπ₯βπ sinβ‘π₯ γ )+(π cosβ‘γπ₯+π sinβ‘π₯ γ ) =0 = R.H.S β΄ Hence Verified