The document provides the steps to solve a first order linear differential equation of the form dy/dx + P(x)y = Q(x). It gives the standard form, explains how to find the integrating factor, and shows the work to solve the specific equation dy/dx + ysecx = tanx. The solution is found to be y = 1 - (x-c)/(secx+tanx).

1. Manash Kumar Mondal

2. Solve the following first order linear
differential equation.
𝒅𝒚
𝒅𝒙
+ ysecx = tanx

3. Knowledge
Standard Form:
𝒅𝒚
𝒅𝒙
+ 𝑷 𝒙 . 𝒚 = 𝑸(𝒙)
Integrating Factor:
𝒆 𝑷 𝒙 𝒅𝒙
Solution :: 𝒚. 𝑰. 𝑭 = 𝑸 𝒙 . 𝑰. 𝑭 𝒅𝒙 + 𝒄

4. Solution
Given, P(x)= secx
𝒅𝒚
𝒅𝒙
+ y. secx = tanx
Q(x)= tanx
# I.F = 𝐞 𝐬𝐞𝐜𝐱 𝐝𝐱
= 𝒆𝒍𝒏 𝒔𝒆𝒄𝒙+𝒕𝒂𝒏𝒙
= secx + tanx

5. ∴ The solution is
𝒚. 𝑰. 𝑭 = 𝑸 𝒙 . 𝑰. 𝑭 𝒅𝒙 + 𝑪
y(secx +tanx) = 𝒕𝒂𝒏𝒙. 𝒔𝒆𝒄𝒙 + 𝒕𝒂𝒏𝒙 𝒅𝒙 + 𝑪
= (𝒕𝒂𝒏𝒙. 𝒔𝒆𝒄𝒙 + 𝒕𝒂𝒏𝒙. 𝒕𝒂𝒏𝒙)𝒅𝒙 + 𝑪
= secx + tanx – x + C
∴ y = 1 – (x-c)/(secx+tanx)
(Ans.)

6. Thank You 