Question

The general solution of differential equation `dx/dy` = cos (x + y) is ______.

पर्याय

• `tan((x + y)/2)` = y + c

• `tan((x + y)/c)` = x + c

• `cot((x + y)/2)` = y + c

• `cot((x + y)/2)` = x + c

Answer 1

The general solution of differential equation `dx/dy` = cos (x + y) is `underlinebb(tan((x + y)/2) = y + c)`.

Explanation:

`dx/dy` = cos (x + y)

`\implies dy/dx = 1/(cos(x + y))`

Put x + y = V

Differentiating w.r.t. ‘x’

`1 + dy/dx = (dV)/dx`

`\implies dy/dx = (dV)/dx - 1`

`\implies (dV)/dx - 1 = 1/cosV`

`\implies (dV)/dx = 1/cosV + 1`

`\implies (dV)/dx = (1 + cosV)/cosV`

`\implies cosV/((1 + cosV)) dV` = dx

Integrate both sides, we get :

`int ((1 + cosV) - 1)/(1 + cosV) dV = int dx`

`\implies int[1 - 1/(2cos^2  V/2)]dV = int dx`

`\implies V - 1/2 (tan  V/2)/(1/2)` = x + C₁

`\implies x + y - tan((x + y)/2)` = x + C₁

`\implies tan((x + y)/2)` = y + C  ...[∵ C = –C₁]