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Question
Solve: dy/dx = cos(x + y)
Solution
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`
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