Question

The solution of the differential equation `dy/dx = (x - 2y + 1)/(2(x - 2y))` is ______ 

पर्याय

• (x - 2y)² + 2x = c

• (x - 2y)² + x = c

• (x - 2y) + 2x² = c 

• (x - 2y) + x² = c 

Answer 1

The solution of the differential equation `dy/dx = (x - 2y + 1)/(2(x - 2y))` is (x - 2y)² + 2x = c.

Explanation:

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

Put x - 2y = v ...................(ii)

⇒ `1 - 2dy/dx = (dv)/dx`

⇒ `dy/dx = 1/2(1 -  (dv)/dx)` .................(iii)

Substituting (ii) and (iii) in (i), we get

`1/2(1 - (dv)/dx) = (v + 1)/(2v)`

⇒ `(dv)/dx = -1/v`

Integrating on both sides, we get

`intv dv = -intdx + c_1`

⇒ `v^2/2 = -x + c_1`

⇒ (x - 2y)² = -2x + 2c₁

⇒ (x - 2y)² + 2x = c, where c = 2c₁