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Example 40 - If y = 3e2x + 2e3x, prove d2y/dx2 - 5 dy/dx

Example 40 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Example 40 - Chapter 5 Class 12 Continuity and Differentiability - Part 3
Example 40 - Chapter 5 Class 12 Continuity and Differentiability - Part 4

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Example 40 If y = 3e2x + 2e3x, prove that 𝑑2𝑦/𝑑π‘₯2 βˆ’ 5 𝑑𝑦/𝑑π‘₯ + 6y = 0. Given, 𝑦 = 3𝑒2π‘₯ + 2𝑒3π‘₯ Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑑(3𝑒2π‘₯ + 2𝑒3π‘₯)/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑑(3𝑒 2π‘₯)/𝑑π‘₯ + 𝑑(2𝑒 3π‘₯)/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 3. 𝑒2π‘₯ .𝑑(2π‘₯)/𝑑π‘₯ + 2 .𝑒 3π‘₯ . 𝑑(3π‘₯)/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 3. 𝑒2π‘₯ . 2 + 2 .𝑒 3π‘₯. 3 𝑑𝑦/𝑑π‘₯ = 6𝑒2π‘₯ + 6𝑒3π‘₯ 𝑑𝑦/𝑑π‘₯ = 6 (𝑒2π‘₯ + 𝑒3π‘₯) Now, π’…π’š/𝒅𝒙 = 6 (π’†πŸπ’™ + π’†πŸ‘π’™) Again Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯ (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 = (𝑑 (6(𝑒2π‘₯" + " 𝑒3π‘₯)))/𝑑π‘₯ (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 = 6 𝑑(𝑒2π‘₯" + " 𝑒3π‘₯)/𝑑π‘₯ (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 = 6(𝑑(𝑒2π‘₯)/𝑑π‘₯ + 𝑑(𝑒3π‘₯)/𝑑π‘₯) (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 = 6(𝑒2π‘₯. 2+𝑒3π‘₯.3) (𝒅^𝟐 π’š)/〖𝒅𝒙〗^𝟐 = 6(πŸπ’†πŸπ’™+πŸ‘π’†πŸ‘π’™) Now we need to prove π’…πŸπ’š/π’…π’™πŸ βˆ’ 5 π’…π’š/𝒅𝒙 + 6y = 0 Solving L.H.S 𝑑2𝑦/𝑑π‘₯2 βˆ’ 5 𝑑𝑦/𝑑π‘₯ + 6y = 6(2𝑒2π‘₯+3𝑒3π‘₯) βˆ’ 5.6 (𝑒2π‘₯+𝑒3π‘₯) + 6(3𝑐2π‘₯+2𝑒3π‘₯) = 12𝑒2π‘₯ + 18𝑒3π‘₯ βˆ’ 30𝑒2π‘₯ βˆ’ 30𝑒3π‘₯ + 18𝑒2π‘₯ + 12𝑒3π‘₯ = 12𝑒2π‘₯ βˆ’ 30𝑒2π‘₯ + 18𝑒2π‘₯ + 18𝑒3π‘₯ βˆ’ 30𝑒3π‘₯ + 12𝑒3π‘₯ = 30𝑒2π‘₯ βˆ’ 30𝑒2π‘₯ + 30𝑒3π‘₯ βˆ’ 30𝑒3π‘₯ = 0 =RHS Hence proved

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.