Question

If y=(tan⁻¹x)²,prove(1+x²)²y₂+2x(1+x²)y₁=2

Answer 1

Answer:

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Answer 2

Answer:

$(1+x^{2} )^{2} y_{2}+2x(1+x^{2})y_{1}=2$ ... Proved.

Step-by-step explanation:

We are given that, y= (tan¬1 x)²....... (1)

And we have to prove that, $(1+x^{2} )^{2} y_{2}+2x(1+x^{2})y_{1}=2$

{ Here $y_{2}=\frac{d^{2}y}{dx^{2}}$ and $y_{1}=\frac{dy}{dx}$ }

Now, differentiating both sides of (1) with respect to x, we get

$y_{1}=\frac{dy}{dx}$= 2(tan¬1 x)$\frac{1}{1+x^{2} }$

{ We have the formula, $\frac{d}{dx}$(tan¬1 x)=$\frac{1}{1+x^{2} }$ }

⇒$y_{1} (1+x^{2} )$= 2(tan¬1 x) ....... (2)

Again, differentiating both sides of (2) with respect to x, we get

$y_{2}(1+x^{2})+y_{1}(2x)=\frac{2}{1+x^{2} }$

{ We have applied the formula, $\frac{d}{dx}(u.v)=u\frac{dv}{dx}+v\frac{du}{dx}$ , where u and v both are functions of x}

⇒ $(1+x^{2} )^{2} y_{2}+2x(1+x^{2})y_{1}=2$

Hence, proved.