Ex 9.5

Chapter 9 Class 12 Differential Equations
Serial order wise

### Transcript

Ex 9.5, 5 For each of the differential equation given in Exercises 1 to 12, find the general solution : cos^2β‘γπ₯ ππ¦/ππ₯+π¦=π‘πππ₯(0β€π₯<π/2)γ Step 1: Put in form ππ¦/ππ₯ + Py = Q cos2x.ππ¦/ππ₯ + y = tan x Dividing by cos2x, ππ¦/ππ₯ + y.1/πππ 2π₯ = tanβ‘π₯/πππ 2π₯ ππ/ππ + (sec2x)y = sec2x. tan x Step 2: Find P and Q Comparing (1) with ππ¦/ππ₯ + Py = Q P = sec2 x and Q = sec2 x. tan x Step 3 : Find integrating factor, I.F I.F = e^β«1βπππ₯ I.F = e^(β«1βγπ ππ2π₯.ππ₯γ " " ) I.F. = etan x Step 4 : Solution of the equation y Γ I.F. = β«1βγπΓπΌ.πΉ.ππ₯γ+π Putting values, y.etan x = β«1βγπππππ.πππβ‘γπ.γ γ "etan x.dx + C" Let I = β«1βγπππππ.πππβ‘γπ.γ γ "etan x.dx" Putting t = tan x" " β΄ π ππ2π₯.dx = dt Putting values of t & dt in equation β΄ I = β«1βtanβ‘γπ₯."etan x" .(π ππ2π₯.ππ₯)γ I =β«1βγπ.π^π.ππγ I = t β«1βγπ^π‘ ππ‘γ β β«1β[ππ‘/ππ‘ β«1βγπ^π‘ ππ‘γ] ππ‘ I = t."et" β β«1βγ"et" ππ‘γ . Using by parts with β«1βγπ(π‘) π(π‘) ππ‘=π(π‘) β«1βγπ(π‘) ππ₯ ββ«1βγ[π^β² (π‘) β«1βγπ(π‘) ππ₯] ππ₯γγγγ Take f (t) = t & g(t) = π^π‘ I = π"et" β "et" Putting t = tan x I = tan x. etan x β etan x I = etan x ( tan x β 1) Substituting value of I in (2), y etan x = etan x (tan x β 1) + C Dividing by etan x, y = tan x β 1 + C. eβtan x