Question

Solve:

dy/dx = cos(x + y)

Answer 1

Given,

`dy/dx= cos (x + y)` …(i)

Put `x + y = v`        …(ii)

`∴ y = v – x`

`∴ dy/dx=(dv)/dx-1`  …(iii)

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

`(dv)/dx-1=cosv`

`therefore (dv)/dx=1+cosv`

`therefore (dv)/dx=2cos^2(v/2)`

`therefore 1/cos^2(v/2)dv=2dx`

`therefore sec^2(v/2)dv=2dx`

Integrating on both sides, we get

`int sec^2(v/2)dv=2intdx`

`therefore 2tan(v/2)=2x+c'`

`therefore tan(v/2)=x+(c')/2`

`therefore tan((x+y)/2)=x+c`, where `c=(c')/2`