Question:

The general solution of $1+\operatorname{Sin}^{2} x=3 \operatorname{Sin} x \cdot \operatorname{Cos} x, \operatorname{Tan} x \neq \frac{1}{2}$ is ......

Updated On: May 22, 2024
  • $n\pi-\frac{\pi}{4},n\in Z$
  • $n\pi+\frac{\pi}{4},n\in Z$
  • $2n\pi+\frac{\pi}{4},n\in Z$
  • $2n\pi-\frac{\pi}{4},n\in Z$
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The Correct Option is B

Solution and Explanation

$1+\sin ^{2} x=3 \sin x \cdot \cos x, \tan x \neq \frac{1}{2}$ Divided by $\cos ^{2} x$ on both sides, $\frac{1}{\cos ^{2} x}+\frac{\sin ^{2} x}{\cos ^{2} x}=3 \frac{\sin x \cdot \cos x}{\cos x \cdot \cos x}$ $\sec ^{2} x+\tan ^{2} x=3 \tan x$ $1+\tan ^{2} x+\tan ^{2} x=3 \tan x$ $2 \tan ^{2} x-3 \tan x+1=0$ $2 \tan ^{2} x-2 \tan x-\tan x+1=0$ $2 \tan x(\tan x-1)-1(\tan x-1)=0$ $(\tan x-1)(2 \tan x-1)=0$ $\tan x=1, \frac{1}{2}$ We take, $\tan x =1 \left(\because \tan x \neq \frac{1}{2}\right)$ $\tan x =\tan (\pi / 4)$ $x =n \pi+\pi / 4, n \in Z$
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Concepts Used:

Differential Equations

A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.

Orders of a Differential Equation

First Order Differential Equation

The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’

Second-Order Differential Equation

The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.

Types of Differential Equations

Differential equations can be divided into several types namely

  • Ordinary Differential Equations
  • Partial Differential Equations
  • Linear Differential Equations
  • Nonlinear differential equations
  • Homogeneous Differential Equations
  • Nonhomogeneous Differential Equations