Question

Write a particular solution of the differential equation, `dy/dx = y^2/(xy − x^2)`, when x = 1 and y = 1.

Answer 1

`dy/dx = y^2/(xy − x^2)`, when x = 1 and y = 1

Put y = Vx 

On differentiating 

`dy/dx = V + x xx (dV)/dx`

On putting in equation (i)

= `V + x(dV)/dx = (V^2x^2)/(x xx Vx − x^2)`

= `V + x(dV)/(dx) = (V^2x^2)/(x^2(V − 1))`

= `V + x (dV)/dx = V^2/(V − 1)`

= x`(dV)/dx = V^2/(V − 1) − V`

= `(V^2 − V^2 + V)/(V−1)`

= `x(dv)/dx = V/(V−1)`

On solving

= `∫(1 − 1/V)dV = ∫dx/x + C`

= V − log V = log x + log C

= `y/x − log  y /x = log x +log C`

= `y/x − log y + log x = log x + log C`    ...[∵ log `a/b = log a − log b`]

`y/x = log y + log C`

`y/x = log Cy`     ...[∵ log ab = log a + log b]

When x = 1, y = 1

Then `1/1 = log C xx 1`

log C = 1

So, `y/x = log y + 1`