Question

Solve the differential equation: \[ \tan^{-1}(y - x) \, dy = (1 + y^2) \, dx. \]

Hint:

For solving differential equations, always look for a way to separate the variables or use an appropriate substitution to simplify the equation.

Answer 1

Step 1: Separate the Variables.
Rearrange the given equation to separate the variables \( y \) and \( x \): \[ \frac{dy}{dx} = \frac{1 + y^2}{\tan^{-1}(y - x)}. \]
Step 2: Solve the Differential Equation.
This equation can be solved using standard methods such as substitution or separation of variables. The solution process involves integrating both sides of the equation, either by direct integration or using a suitable substitution.
Step 3: Conclusion.
The exact solution requires a series of integration steps. After solving, you will obtain the general solution for \( y \) as a function of \( x \).