Question

Find the general solution of \[ \frac{dy}{dx} = \frac{x - 1}{2 + y}. \]

Hint:

When solving separable differential equations, separate the variables and then integrate both sides to find the general solution.

Answer 1

We are given the differential equation: \[ \frac{dy}{dx} = \frac{x - 1}{2 + y}. \]

Step 1: Rearrange the equation to separate variables.
Rearrange the equation to separate the variables \( x \) and \( y \): \[ (2 + y) \, dy = (x - 1) \, dx. \]

Step 2: Integrate both sides.
Now, integrate both sides: \[ \int (2 + y) \, dy = \int (x - 1) \, dx. \] On the left-hand side, the integral of \( 2 + y \) is: \[ \int (2 + y) \, dy = 2y + \frac{y^2}{2}. \] On the right-hand side, the integral of \( x - 1 \) is: \[ \int (x - 1) \, dx = \frac{x^2}{2} - x. \]

Step 3: Include the constant of integration.
Equating the results of the integrals and adding the constant of integration \( C \) on the right-hand side: \[ 2y + \frac{y^2}{2} = \frac{x^2}{2} - x + C. \] Conclusion: The general solution is: \[ 2y + \frac{y^2}{2} = \frac{x^2}{2} - x + C. \]