Question

Solve the differential equation:
\[ y \, dx + (x - y^2) \, dy = 0 \]

Hint:

Remember: When solving differential equations, always look for the method of separation of variables or apply substitution techniques to simplify the integrals.

Answer 1

Step 1: Rearrange the equation.
Rearrange the given differential equation:
y dx = -(x - y²) dy
Now divide both sides by y(x - y²):
dx / (x - y²) = - dy / y

Step 2: Integrate both sides.
Now, integrate both sides:
∫ dx / (x - y²) = - ∫ dy / y

The left-hand side is the integral of 1 / (x - y²), which is tricky. After simplifying and making appropriate substitutions, we can solve the equation for x and y. But a general solution could be complex and might require special techniques.