---
title: "Solve dy/dx − 2y tan x = y² tan x"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-bernoulli-2y-tan-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-bernoulli-2y-tan-x
site: BCT Engineering
---

# Solve dy/dx − 2y tan x = y² tan x

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

- Asked in: 2068 Chaitra
- Video: [Calculus Help: Bernoulli's Differential Equations - dy/dx-2y tanx=y^2  tan^2⁡x - Techniques](https://www.youtube.com/watch?v=7S5efSyxL7U) — Calculus Physics Chem Accounting Tam Mai Thanh Cao
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-bernoulli-2y-tan-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve the differential equation dy/dx − 2y tan x = y² tan x.

## Short answer

It is Bernoulli with n=2. Divide by y² and substitute z=1/y to get a linear equation. Using integrating factor sec²x and solving yields 1/y = c cos²x - 1/2 sin²x.

## Answer

dy/dx − 2y tan x = y² tan x

This is Bernoulli's equation with n = 2.

#### Reduce to linear form
Divide by y²:

y⁻² dy/dx − 2y⁻¹ tan x = tan x

Put z = 1/y, so dz/dx = −y⁻² dy/dx:

−dz/dx − 2z tan x = tan x → dz/dx + 2 tan x · z = −tan x

#### Integrating factor
IF = e^(2∫tan x dx) = e^(2 log sec x) = sec²x

#### Solution
z sec²x = −∫ tan x sec²x dx = −tan²x/2 + c

z = c cos²x − (1/2) tan²x cos²x = c cos²x − (1/2) sin²x

Replace z = 1/y:

**1/y = c cos²x − ½ sin²x**

or 2 = y(C cos²x − sin²x) with C = 2c.

## Check yourself

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

1. dy/dx - 2y tan x = y^2 tan x is a Bernoulli equation with n =
   A. 1
   B. -2
   C. 2
   D. 0
   Answer: C. The power of y on the right is 2. (Key point: Type)

2. The substitution to linearise is:
   A. z = 1/y
   B. z = y/x
   C. z = ln y
   D. z = y^2
   Answer: A. Divide by y^2 and let z = y^(1-n) = 1/y. (Key point: Substitution)

3. The linear equation in z is:
   A. dz/dx + 2 tan x z = tan x
   B. dz/dx + 2 tan x z = -tan x
   C. dz/dx - 2 tan x z = tan x
   D. dz/dx + tan x z = -2 tan x
   Answer: B. -dz/dx - 2z tan x = tan x. (Key point: Linear equation)

4. The integrating factor is:
   A. sec x
   B. tan x
   C. cos^2 x
   D. sec^2 x
   Answer: D. e^(2 log sec x) = sec^2 x. (Key point: Integrating factor)

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