Examples

Chapter 9 Class 12 Differential Equations
Serial order wise

### Transcript

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 Now, (π^2 π¦)/(ππ₯^2 )=π/ππ₯ (ππ¦/ππ₯) (π^π π)/(ππ^π ) =π/ππ (βπ ππππ+π ππππ) (π^2 π¦)/(ππ₯^2 ) =βπ π(sinβ‘π₯ )/ππ₯+π (π(cosβ‘π₯))/ππ₯ (π^2 π¦)/(ππ₯^2 ) =βπ(cosβ‘π₯ )+π(βsinβ‘π₯) (π^2 π¦)/(ππ₯^2 ) =βπ πππ β‘γπ₯βπ π ππβ‘π₯ γ (π^π π)/(ππ^π ) =β(π πππβ‘γπ+π πππβ‘π γ ) Putting y = π cosβ‘γπ₯+π sinβ‘γπ₯ γ γ (π^π π)/(ππ^π )=βπ (π^π π)/(ππ^π )+π=π β΄ Hence Verified