Question:
The solution of $\left(x+y\right)^{2} \left(x \frac{dy}{dx}+y\right)=xy \left(1 +\frac{dy}{dx}\right) $ is
The solution of $\left(x+y\right)^{2} \left(x \frac{dy}{dx}+y\right)=xy \left(1 +\frac{dy}{dx}\right) $ is
Updated On: May 27, 2023
- $\log\left(xy\right) =- \frac{1}{x+y} +C$
- $\log\left(\frac{x}{y}\right) =- \frac{1}{x+y} +C$
- $\log\left(xy\right) =- \frac{1}{x - y} +C$
- $None \,of\, these$
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The Correct Option is A
Solution and Explanation
Given D.E is
$\left(x+y\right)^{2} \left(x \frac{dy}{dx} +y\right) =xy \left(1 + \frac{dy}{dx}\right) $
$\Rightarrow \left(xy\right)^{-1} \left(x \frac{dy}{dx}+y\right)= \left(x+y\right)^{-2} \left(1+\frac{dy}{dx}\right)$
$\Rightarrow \int\left(xy\right)^{-1} \left(x \frac{dy}{dx} +y\right) = \int \left(x +y\right)^{-2} \left(1 + \frac{dy}{dx} \right)$
Using integral $\int\left(f\left(x\right)\right)^{n} f'\left(x\right)dx = \frac{\left(f\left(x\right)\right)^{n+1}}{n+1} $ and $\int\frac{f'\left(x\right)}{f\left(x\right)} dx = \log\left(f\left(x\right)\right)+C $
$\Rightarrow \log\left(xy\right) =\frac{\left(x+y\right)^{-1}}{-1} +C $
$\Rightarrow \log\left(xy\right) =\frac{-1}{x+y} +C$
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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.