Question

Solve: \[ 1 + \frac{dy}{dx} = {cosec}(x + y); \quad {put } x + y = u. \]

Hint:

Use substitution to simplify differential equations of the form \( f(x + y) \).

Answer 1

Step 1: Substituting \( u = x + y \)
\[ du = dx + dy \Rightarrow \frac{du}{dx} = 1 + \frac{dy}{dx}. \] Step 2: Rewrite the Equation
\[ \frac{du}{dx} = {cosec } u. \] Step 3: Solve the Differential Equation
\[ \int \sin u \,du = \int dx. \] \[ - \cos u = x + C. \] \[ - \cos(x + y) = x + C. \]