Question

Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`

Answer 1

Given, `x + y (dy)/(dx) = sec(x^2 + y^2)`  .....(i)

Put x² + y² = t

Differentiating w.r.t. x, we get

`2x + 2y (dy)/(dx) = (dt)/(dx)`

⇒ `x + y (dy)/(dx) = 1/2 (dt)/(dx)`

Put this value in (i), we get

`1/2 (dt)/(dx)` = sec t

⇒ `(dt)/sec t` = 2 dx

Integrating on both sides, we get

`int cost  dt = 2 int  dx`

⇒ sin t = 2x + c

⇒ sin(x² + y²) = 2x + C   ......(∵ t = x² + y²) 

is the required solution.