Question

If x=tan(1/a logy) Show that (1+x2)d2y/dx2 + (2x-a)dy/dx = 0

Answer 1

HELLO DEAR,

GIVEN:-

x = $\bold{tan(\frac{1}{a}logy)}$

so, tan^{-1}x = 1/a logy

differentiating both with respect to x.

we get,

$\bold{\Rightarrow \frac{1}{1 + x^2} = \frac{1}{a} * \frac{1}{y} \frac{dy}{dx}}$

$\bold{\Rightarrow \frac{a}{1 + x^2} = \frac{1}{y} \frac{dy}{dx}}$

$\bold{\Rightarrow (1 + x^2)\frac{dy}{dx} = ay}$

again, differentiating with respect to x.

$\bold{\Rightarrow (1 + x^2)\frac{d^2y}{dx^2} + (2x)\frac{dy}{dx} = a\frac{dy}{dx}}$

$\bold{\Rightarrow (1 + x^2)\frac{d^2y}{dx^2} + (2x - a)\frac{dy}{dx} = 0}$

$\bold{\Large{HENCE, \,\,PROVED}}$

I HOPE ITS HELP YOU DEAR,

THANKS,