Question

$\frac{dy}{dx}tan(y - x.\frac{dy}{dx})$ ആണെങ്കിൽ y

• A. y-cx = tan cx
• B. y + cx = tan c
• C. $$y + cx = tan^{-1}c$$
• D. $$y-cx=tan{-1}c$$

Answer 1

Answer:

$$y-cx=tan{-1}c$$

Given differential equation:
$\frac{dy}{dx} = \tan\left(y - x\frac{dy}{dx}\right)$

Let $( p = \frac{dy}{dx} )$

Then equation becomes:

$p = \tan(y - xp)$

Take inverse tan

$y - xp = \tan^{-1}(p)$

Rearrange

$y = xp + \tan^{-1}(p)$

This is a Clairaut’s equation of the form:
$y = xp + f(p)$

General solution

Replace (p) by constant (c):

$y = cx + \tan^{-1}(c)$

Final Answer:
$\boxed{y = cx + \tan^{-1}(c)}$