Question:
The solution of
$\frac{dy}{dx} = \frac{y}{x}+\tan \frac{y}{x}$ is
The solution of
$\frac{dy}{dx} = \frac{y}{x}+\tan \frac{y}{x}$ is
Updated On: Apr 26, 2024
- x = c sin (y/x)
- x = c sin (xy)
- y = c sin (y/x)
- xy = c sin (x/y)
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The Correct Option is A
Solution and Explanation
Given, $\frac{d y}{d x}=\frac{y}{x}+\tan \frac{y}{x}$
Put $y=v x \Rightarrow \frac{d y}{d x}=x \frac{d v}{d x}+v$
$\therefore x \frac{d v}{d x}+v=v+\tan v$
$\Rightarrow \cot v\, d v=\frac{1}{x} d x$
On integrating both sides, we get
$\Rightarrow \log c+\log \sin v =\log x \\ c \sin v =x $
$\Rightarrow x=c \sin \left(\frac{y}{x}\right)$
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Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.