Ex 9.5, 4 - Chapter 9 Class 12 Differential Equations

Transcript

Ex 9.5, 4 For each of the differential equation given in Exercises 1 to 12, find the general solution : 𝑑𝑦/𝑑𝑥+(sec⁡𝑥 )𝑦=𝑡𝑎𝑛𝑥(0≤𝑥<𝜋/2) Differential equation is of the form 𝑑𝑦/𝑑𝑥 + Py = Q where P = sec x and Q = tan x Finding integrating factor, IF = 𝑒^∫1▒〖𝑝 𝑑𝑥〗 IF = e^∫1▒sec⁡〖𝑥 𝑑𝑥〗 IF = 〖e^𝑙𝑜𝑔〗^|sec⁡〖𝑥 + tan⁡𝑥 〗 | I.F = sec x + tan x Solution is y (IF) = ∫1▒(𝑄×𝐼.𝐹) 𝑑𝑥+𝑐 y (sec x + tan x) = ∫1▒〖𝐭𝐚𝐧⁡𝒙 (𝒔𝒆𝒄⁡𝒙+𝐭𝐚𝐧⁡〖𝒙)〗 〗+𝒄 y (sec x + tan x) = ∫1▒〖tan⁡𝑥 sec⁡𝑥 〗 𝑑𝑥+∫1▒〖tan^2⁡𝑥 𝑑𝑥+𝐶〗 y (sec x + tan x) = sec x + ∫1▒〖(sec^2⁡𝑥−1)〗 dx + c y (sec x + tan x) = sec x + 𝐭𝐚𝐧⁡𝒙 − x + c