Question:
If $\log y= m \tan^{-1} x,$ then
If $\log y= m \tan^{-1} x,$ then
Updated On: May 12, 2024
- $(1 + x^2)y^2 + (2x + m)y_1 = 0$
- $(1 + x^2)y^2 + (2x - m)y_1 = 0$
- $(1 + x^2)y^2 - (2x + m)y_1 = 0$
- $(1 + x^2)y^2 - (2x - m)y_1 = 0$
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The Correct Option is B
Solution and Explanation
We have, $\log y= m \tan^{-1} x$
Differentating w.r.t. x, we get
$\frac{1}{y} \frac{dy}{dx} = \frac{m}{1+x^{2}} $
or , $y_{1} \left(1+x^{2}\right) =my$
Again differentiating w.r.t, x, we get
$y_{2}\left(1+x^{2}\right)+y_{1}\left(2x\right)=my_{1} $
$\Rightarrow \left(1+x^{2}\right) y_{2} +\left(2x -m\right)y_{1} = 0$
Differentating w.r.t. x, we get
$\frac{1}{y} \frac{dy}{dx} = \frac{m}{1+x^{2}} $
or , $y_{1} \left(1+x^{2}\right) =my$
Again differentiating w.r.t, x, we get
$y_{2}\left(1+x^{2}\right)+y_{1}\left(2x\right)=my_{1} $
$\Rightarrow \left(1+x^{2}\right) y_{2} +\left(2x -m\right)y_{1} = 0$
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.