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

Question

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

Ankush Jain · Accepted Answer

Given, $x = \tan â¡ (\frac{1}{a} \log â¡ â€‰ y) â‡’ \tan^{âˆ’ 1} x = \frac{1}{a} \log â¡ â€‰ y â‡’ a â¢ {â€‰ â€‰ \tan}^{âˆ’ 1} x = \log â¡ â€‰ y$ Differentiating w r t x, we get $a â¢ \frac{d}{d x} ({â€‰ â€‰ \tan}^{âˆ’ 1} x) = \frac{d}{d x} (\log â¡ â€‰ y) â‡’ \frac{a}{1 + x^{2}} = \frac{1}{y} \frac{d y}{d x} â‡’ (1 + x^{2}) \frac{d y}{d x} = a y$ Again, differentiating w r t x, we get $\frac{d}{d x} [(1 + x^{2}) \frac{d y}{d x}] = a \frac{d}{d x} (y) â‡’ (1 + x^{2}) \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} (2 x) = a \frac{d y}{d x} â‡’ (1 + x^{2}) \frac{d^{2} y}{d x^{2}} + (2 x âˆ’ a) \frac{d y}{d x} = 0$ [Hence proved]