---
title: "Solve dy/dx + y tan x = sec x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-linear-y-tan-x-sec-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-linear-y-tan-x-sec-x
site: BCT Engineering
---

# Solve dy/dx + y tan x = sec x

Mathematics I · Chapter 4: Ordinary Differential Equations and their Applications

- Asked in: 2073 Chaitra
- Video: [dy/dx+ytanx=secx #LinearEquation L607 @MathsPulseChinnaiahKalpana](https://www.youtube.com/watch?v=1Xe_Zt5iuFo) — Maths Pulse - Chinnaiah Kalpana
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-linear-y-tan-x-sec-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve: dy/dx + y tan x = sec x.

## Short answer

This is linear with integrating factor sec x. Multiplying and integrating gives y sec x = tan x + c, so y = sin x + c cos x.

## Answer

dy/dx + y tan x = sec x

This is a linear first-order equation, P = tan x, Q = sec x.

#### Integrating factor
IF = e^(∫tan x dx) = e^(log sec x) = sec x

#### Solution
y · sec x = ∫ sec x · sec x dx = ∫ sec²x dx = tan x + c

Multiply by cos x:

**y = sin x + c cos x**

(Check: y' + y tan x = cos x − c sin x + sin x tan x + c sin x = cos x + sin²x/cos x = 1/cos x = sec x ✓.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For dy/dx + y tan x = sec x, the IF is:
   A. cos x
   B. sin x
   C. sec x
   D. tan x
   Answer: C. e^(integral tan x dx) = e^(log sec x) = sec x. (Key point: Integrating factor)

2. The right-hand integral is:
   A. integral of sec^2 x dx = tan x + c
   B. integral of sec x dx
   C. integral of tan x dx
   D. sec x tan x + c
   Answer: A. IF × Q = sec x · sec x. (Key point: Right-hand integral)

3. The solution is:
   A. y = cos x + c sin x
   B. y = tan x + c
   C. y = sin x + c sec x
   D. y = sin x + c cos x
   Answer: D. Multiply y sec x = tan x + c by cos x. (Key point: Final solution)

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