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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 37 If y = 3e2x + 2e3x, prove that 𝑑2𝑦/𝑑𝑥2 − 5 𝑑𝑦/𝑑𝑥 + 6y = 0. Given, 𝑦 = 3𝑒2𝑥 + 2𝑒3𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦/𝑑𝑥 = 𝑑(3𝑒2𝑥 + 2𝑒3𝑥)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑑(3𝑒 2𝑥)/𝑑𝑥 + 𝑑(2𝑒 3𝑥)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 3. 𝑒2𝑥 .𝑑(2𝑥)/𝑑𝑥 + 2 .𝑒 3𝑥 . 𝑑(3𝑥)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 3. 𝑒2𝑥 . 2 + 2 .𝑒 3𝑥. 3 𝑑𝑦/𝑑𝑥 = 6𝑒2𝑥 + 6𝑒3𝑥 𝑑𝑦/𝑑𝑥 = 6 (𝑒2𝑥 + 𝑒3𝑥) Now, 𝒅𝒚/𝒅𝒙 = 6 (𝒆𝟐𝒙 + 𝒆𝟑𝒙) Again Differentiating 𝑤.𝑟.𝑡.𝑥 (𝑑^2 𝑦)/〖𝑑𝑥〗^2 = (𝑑 (6(𝑒2𝑥" + " 𝑒3𝑥)))/𝑑𝑥 (𝑑^2 𝑦)/〖𝑑𝑥〗^2 = 6 𝑑(𝑒2𝑥" + " 𝑒3𝑥)/𝑑𝑥 (𝑑^2 𝑦)/〖𝑑𝑥〗^2 = 6(𝑑(𝑒2𝑥)/𝑑𝑥 + 𝑑(𝑒3𝑥)/𝑑𝑥) (𝑑^2 𝑦)/〖𝑑𝑥〗^2 = 6(𝑒2𝑥. 2+𝑒3𝑥.3) (𝒅^𝟐 𝒚)/〖𝒅𝒙〗^𝟐 = 6(𝟐𝒆𝟐𝒙+𝟑𝒆𝟑𝒙) Now we need to prove 𝒅𝟐𝒚/𝒅𝒙𝟐 − 5 𝒅𝒚/𝒅𝒙 + 6y = 0 Solving L.H.S 𝑑2𝑦/𝑑𝑥2 − 5 𝑑𝑦/𝑑𝑥 + 6y = 6(2𝑒2𝑥+3𝑒3𝑥) − 5.6 (𝑒2𝑥+𝑒3𝑥) + 6(3𝑐2𝑥+2𝑒3𝑥) = 12𝑒2𝑥 + 18𝑒3𝑥 − 30𝑒2𝑥 − 30𝑒3𝑥 + 18𝑒2𝑥 + 12𝑒3𝑥 = 12𝑒2𝑥 − 30𝑒2𝑥 + 18𝑒2𝑥 + 18𝑒3𝑥 − 30𝑒3𝑥 + 12𝑒3𝑥 = 30𝑒2𝑥 − 30𝑒2𝑥 + 30𝑒3𝑥 − 30𝑒3𝑥 = 0 =RHS Hence proved

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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.