Question

Solve the differential equation \( \frac{dy}{dx} = \frac{x^2 - 1}{y^2 + 1} \).

Hint:

The first step in solving a first-order differential equation should always be to check if it's separable. If the equation can be written in the form \( f(y)dy = g(x)dx \), this method is usually the simplest.

Answer 1

Step 1: Understanding the Concept:
This is a first-order differential equation. We can see that the variables can be separated, with all y-terms on one side and all x-terms on the other. This method is called separation of variables.
Step 2: Key Formula or Approach:
1. Rearrange the equation so that all terms involving y and dy are on one side, and all terms involving x and dx are on the other.
2. Integrate both sides of the equation with respect to their respective variables.
3. The result is the general solution to the differential equation.
Step 3: Detailed Explanation or Calculation:
The given equation is:
\[ \frac{dy}{dx} = \frac{x^2 - 1}{y^2 + 1} \] Separate the variables by multiplying both sides by \( (y^2 + 1)dx \):
\[ (y^2 + 1) dy = (x^2 - 1) dx \] Now, integrate both sides:
\[ \int (y^2 + 1) dy = \int (x^2 - 1) dx \] \[ \frac{y^3}{3} + y = \frac{x^3}{3} - x + C \] where C is the constant of integration.
This is the general solution in implicit form. We can also write it as:
\[ y^3 + 3y = x^3 - 3x + 3C \] Let \( K = 3C \), another constant.
\[ y^3 + 3y = x^3 - 3x + K \] Step 4: Final Answer:
The general solution to the differential equation is \( \frac{y^3}{3} + y = \frac{x^3}{3} - x + C \).