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Ex 9.6, 5 - Find general solution: cos2 x dy/dx + y = tan x - Solving Linear differential equations - Equation given

Ex 9.6, 5 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.6, 5 - Chapter 9 Class 12 Differential Equations - Part 3
Ex 9.6, 5 - Chapter 9 Class 12 Differential Equations - Part 4

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Transcript

Ex 9.6, 5 For each of the differential equation given in Exercises 1 to 12, find the general solution : cos﷮2﷯﷮𝑥 𝑑𝑦﷮𝑑𝑥﷯+𝑦=𝑡𝑎𝑛𝑥 0≤𝑥< 𝜋﷮2﷯﷯﷯ Step 1: Put in form 𝑑𝑦﷮𝑑𝑥﷯ + Py = Q cos2x. 𝑑𝑦﷮𝑑𝑥﷯ + y = tan x Dividing by cos2x, 𝑑𝑦﷮𝑑𝑥﷯ + y. 1﷮𝑐𝑜𝑠2𝑥﷯ = tan﷮𝑥﷯﷮𝑐𝑜𝑠2𝑥﷯ ⇒ 𝑑𝑦﷮𝑑𝑥﷯ + (sec2x)y = sec2x. tan x Step 2: Find P and Q Comparing (1) with 𝑑𝑦﷮𝑑𝑥﷯ + Py = Q P = sec2 x and Q = sec2 x. tan x Step 3 : Find integrating factor, I.F I.F = e﷮ ﷮﷮𝑝𝑑𝑥﷯﷯ I.F = e﷮ ﷮﷮𝑠𝑒𝑐2𝑥.𝑑𝑥﷯ ﷯ I.F. = etan x Step 4 : Solution of the equation y × I.F. = ﷮﷮𝑄×𝐼.𝐹.𝑑𝑥﷯+𝑐 Putting values, y.etan x = ﷮﷮𝑠𝑒𝑐2𝑥. tan﷮𝑥.﷯﷯etan x.dx + C Let I = ﷮﷮𝑠𝑒𝑐2𝑥. tan﷮𝑥.﷯﷯etan x.dx Putting t = tan x ⇒ 𝑠𝑒𝑐2𝑥.dx = dt Putting values of t & dt in equation ∴ I = ﷮﷮ tan﷮𝑥.etan x.(𝑠𝑒𝑐2𝑥.𝑑𝑥)﷯﷯ I = ﷮﷮𝑡. 𝑒﷮𝑡﷯.𝑑𝑡﷯ I = t ﷮﷮ 𝑒﷮𝑡﷯𝑑𝑡﷯ − ﷮﷮ 𝑑𝑡﷮𝑑𝑡﷯ ﷮﷮ 𝑒﷮𝑡﷯𝑑𝑡﷯﷯﷯𝑑𝑡 I = t.et − ﷮﷮et 𝑑𝑡﷯ I = 𝑡et − et . Putting t = tan x I = tan x. etan x – etan x I = etan x ( tan x − 1) Substituting value of I in (2), y etan x = etan x (tan x − 1) + C Dividing by etan x, y = tan x − 1 + C. e–tan x

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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.