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Misc 2 For each of the exercise given below , verify that the given function (π‘–π‘šπ‘π‘™π‘–π‘π‘–π‘‘ π‘œπ‘Ÿ 𝑒π‘₯𝑝𝑙𝑖𝑐𝑖𝑑) is a solution of the corresponding differential equation . (iii) 𝑦=π‘₯ sin⁑3π‘₯ : (𝑑^2 𝑦)/(𝑑π‘₯^2 )+9π‘¦βˆ’6 cos⁑〖3π‘₯=0γ€— 𝑦=π‘₯ sin⁑3π‘₯ Differentiating w.r.t x 𝑦^β€²=(π‘₯ 𝑠𝑖𝑛⁑3π‘₯ )^β€² 𝑦^β€²=π‘₯^β€² sin⁑3π‘₯+π‘₯(sin⁑3π‘₯)β€² 𝑦^β€²=sin⁑3π‘₯+π‘₯Γ—3 cos⁑3π‘₯ π’š^β€²=π’”π’Šπ’β‘πŸ‘π’™+πŸ‘π’™ π’„π’π’”β‘πŸ‘π’™ Differentiating again w.r.t. x 𝑦^β€²β€²=(sin⁑3π‘₯ )^β€²+(3π‘₯ cos⁑3π‘₯ )^β€² 𝑦^β€²β€²=3 cos⁑3π‘₯+γ€–3(π‘₯)γ€—^β€² cos⁑3π‘₯+3π‘₯ (cos⁑3π‘₯ )^β€² 𝑦^β€²β€²=3 cos⁑3π‘₯+3 cos⁑3π‘₯+3π‘₯(βˆ’3 sin⁑3π‘₯) π’š^β€²β€²=πŸ” π’„π’π’”β‘πŸ‘π’™βˆ’πŸ—π’™ π’”π’Šπ’β‘πŸ‘π’™ Putting 𝑦=π‘₯ sin⁑3π‘₯ 𝑦^β€²β€²=6 cos⁑3π‘₯βˆ’9𝑦 π’š^β€²β€²+πŸ—π’šβˆ’πŸ” π’„π’π’”β‘πŸ‘π’™=𝟎 Thus, Given Function is a solution of the Differential Equation

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.