Question

If x=tan(1/a logy), show that (1+x^2)d^2y/dx^2+(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,