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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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.