Question

Solve the differential equation `sin^(-1) (dy/dx) = x + y`

Answer 1

Put x + y = t

`1 + dy/dx = dt/dx`

`dy/dx = dt/dx - 1`

Now, `dy/dx = sin (x + y)`

`dt/dx - 1 = sin t`

`dt/dx = 1 +  sin t`

`int dt/(1+ sint) =  int dx`

`int (1 -sint)/(1-sin^2 t)dt = x + c`

`int  (1- sint)/cos^2t dt = x + c`

`int sec^2 t dt  - int tant sec t dt =  x + c`

`tan t - sect  =  x + c`

`tan(x + y) -  sec(x + y) = x + c`