Question

If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x²dy = (2xy + y²)dx, then `f(1/2)` is equal to ______.

Options

• `1/(1 + log_e2)`

• `1/(1 - log_e2)`

• 1 + log_e2

• `(-1)/(1 + log_e2)`

Answer 1

If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x²dy = (2xy + y²)dx, then `f(1/2)` is equal to `underlinebb(1/(1 + log_e2))`.

Explanation:

`(dy)/(dx) = (2xy + y^2)/(2x^2)` 

It is homogeneous differential equation.

∴ Put y = vx

`\implies v + x(dv)/(dx) = v + v^2/2`

`\implies int2(dv)/v^2 = int(dx)/x`

`\implies (-2)/v` = log_e x + c

`\implies (-2x)/y` = log_e x + c

Put x = 1, y = 2, we get c = –1

`\implies (-2x)/y` = log_e x –1

Hence, put x = `1/2` `\implies` y = `1/(1 + log_e 2)`