Example 19 - Chapter 9 Class 12 Differential Equations
Last updated at April 16, 2024 by Teachoo
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Question 1 Deleted for CBSE Board 2025 Exams
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Last updated at April 16, 2024 by Teachoo
Example 19 Verify that the function π¦=π1 π^ππ₯ cosβ‘γππ₯+π2 π^ππ₯ sinβ‘ππ₯ γ , π€βπππ π1 , π2 are arbitrary constants is a solution of the differential equation (π^2 π¦)/(ππ₯^2 )β2π ππ¦/ππ₯+(π^2+π^2 )π¦=0 π¦=π1 π^ππ₯ cosβ‘γππ₯+π2 π^ππ₯ sinβ‘γππ₯, γ γ π =π^ππ (ππ πππ ππ+ ππ πππβ‘ππ) Differentiating w.r.t. x π¦^β²=(π^ππ₯ )^β² (π_1 cosβ‘ππ₯+π_2 sinβ‘ππ₯ )+π^ππ₯ (π_1 cosβ‘ππ₯+π_2 sinβ‘ππ₯ )^β² π¦^β²=ππ^ππ₯ (π_1 cosβ‘ππ₯+π_2 sinβ‘ππ₯ )+π^ππ₯ (γβπγ_1 π sinβ‘ππ₯+π_2 π cosβ‘ππ₯ ) π¦^β²=ππ^ππ₯ (π_1 cosβ‘ππ₯+π_2 sinβ‘ππ₯ )+ππ^ππ₯ (γβπγ_1 sinβ‘ππ₯+π_2 cosβ‘ππ₯ ) Putting π¦ =π^ππ₯ (π1 πππ ππ₯+ π2 sinβ‘ππ₯) π¦^β²=ππ¦+ππ^ππ₯ (γβπγ_1 sinβ‘ππ₯+π_2 cosβ‘ππ₯ ) π^β²βππ=ππ^ππ (γβπγ_π πππβ‘ππ+π_π πππβ‘ππ ) Differentiating again w.r.t x π¦^β²β²βππ¦^β²=(ππ^ππ₯ )^β² (γβπγ_1 sinβ‘ππ₯+π_2 cosβ‘ππ₯ )+ππ^ππ₯ (γβπγ_1 sinβ‘ππ₯+π_2 cosβ‘ππ₯ )^β² π¦^β²β²βππ¦^β²=πππ^ππ₯ (γβπγ_1 sinβ‘ππ₯+π_2 cosβ‘ππ₯ )+ππ^ππ₯ (γβπγ_1 π cosβ‘ππ₯βπ_2 π sinβ‘ππ₯ ) π¦^β²β²βππ¦^β²=πππ^ππ₯ (γβπγ_1 sinβ‘ππ₯+π_2 cosβ‘ππ₯ )βπ^2 π^ππ (π_π πππβ‘ππ+π_π πππβ‘ππ ) Putting π¦ =π^ππ₯ (π1 πππ ππ₯+ π2 sinβ‘ππ₯) π¦^β²β²βππ¦^β²=πππ^ππ (γβπγ_π πππβ‘ππ+π_π πππβ‘ππ )βπ^2 π¦ "Putting" π¦^β²βππ¦=ππ^ππ₯ (γβπγ_1 π ππβ‘ππ₯+π_2 πππ β‘ππ₯ ) π¦^β²β²βππ¦^β²=π(π¦^β²βππ¦)βπ^2 π¦ π¦^β²β²βππ¦^β²=ππ¦^β²βπ^2 π¦βπ^2 π¦ π¦^β²β²βππ¦^β²βππ¦^β²+π^2 π¦+π^2 π¦=0 π^β²β²βπππ^β²+(π^π+π^π)π=π Hence verified