Question

If the solution curve y = y(x) of the differential equation y²dx + (x² – xy + y²)dy = 0, which passes through the point (1, 1) and intersects the line y = `sqrt(3)  x` at the point `(α, sqrt(3) α)`, then value of `log_e (sqrt(3)α)` is equal to ______.

Options

• `π/3`

• `π/2`

• `π/12`

• `π/6`

Answer 1

If the solution curve y = y(x) of the differential equation y²dx + (x² – xy + y²)dy = 0, which passes through the point (1, 1) and intersects the line y = `sqrt(3)  x` at the point `(α, sqrt(3) α)`, then value of `log_e (sqrt(3)α)` is equal to `underlinebb(π/12)`.

Explanation:

Given differential equation is

y²dx – xy dy = – (x² + y²) dy

`(dy)/(dx) = y^2/(xy - x^2 - y^2)`

Divide by x² in numerator and denominator.

`(dy)/(dx) = (y/x)^2/((y/x) - 1 - (y/x)^2`  ...(i)

Let `y/x` = v, y = vx; Differentiate w.r.t. x.

`(dy)/(dx) = v + (dv)/(dx).x`  ...[Put in equation (i)]

`v + (dv)/(dx)x = v^2/(v - 1 - v^2) \implies (v - 1 - v^2)/(v(1 + v^2)).dv = int (dx)/x`

`\implies tan^-1(y/x)` = log_e y + C

Put x = 1, y = 1, then C = `π/4`

Now, satisfy point  `(α, sqrt(3) α) \implies π/3 = log_e (sqrt(3)α) + π/4`

∴ `log_e (sqrt(3)α) = π/12`