If $y = y ( x ), x \in\left(0, \frac{\pi}{2}\right)$ be the solution curve of the differential equation$ \left(\sin ^2 2 x\right) \frac{d y}{d x}+\left(8 \sin ^2 2 x+2 \sin 4 x\right) y$ $ =2 e^{-4 x}(2 \sin 2 x+\cos 2 x), \text { with } y\left(\frac{\pi}{4}\right)=e^{-\pi},$ then $y\left(\frac{\pi}{6}\right)$ is equal to:
$\frac{2}{\sqrt{3}} e ^{-\frac{2 \pi }{ 3}}$
$\frac{2}{\sqrt{3}} e ^{\frac{2 \pi }{ 3}}$
$\frac{1}{\sqrt{3}} e ^{-\frac{2 \pi }{ 3}}$
$\frac{1}{\sqrt{3}} e ^{\frac{2 \pi }{ 3}}$
The Correct Option is A
Solution and Explanation
The correct answer is (A) : \(\frac{2}{\sqrt3}e^{-\frac{2\pi}{3}}\)
Given differential equation
\(\frac{dy}{dx}+(8+4\cot2x)y=\frac{2e^{-4x}}{\sin^22x}(2\sin2x+\cos2x)\)
\(I.F=\int_{e}(8+4\cot2x)dy=e^{8x+2log_e(\sin2x)}=e^{8x}.\sin^22x\)
Solution is
\(y(e^{8x}.\sin^22x)=\int2e^{4x}(2\sin2x+\cos2x)dx+C\)
\(\therefore y(x)=\frac{e^{-4x}}{\sin2x}\)
\(\therefore y(\frac{\pi}{6})=\frac{e^{-4\frac{\pi}{6}}}{\sin(2.\frac{\pi}{6})}\)
\(=\frac{2}{\sqrt3}e^{-\frac{2\pi}{3}}\)
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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