Question

Find the general solution of differential equation \( ydx + (x - y^2)dy = 0 \).

Hint:

{Key Points:}

• Recognize exact differentials: \( d(xy) = x\,dy + y\,dx \)
• Rearrange equation to identify such forms
• Integrate both sides after recognizing exact differential

Answer 1

Step 1: Identify the form of differential equation The given equation is: \[ y \, dx + (x - y^2) \, dy = 0 \] This can be written as: \[ y \, dx + x \, dy - y^2 \, dy = 0 \]
Step 2: Rearrange terms \[ y \, dx + x \, dy = y^2 \, dy \]
Step 3: Recognize the left-hand side The left-hand side \( y \, dx + x \, dy \) is the differential of \( xy \): \[ d(xy) = x \, dy + y \, dx \] Therefore: \[ d(xy) = y^2 \, dy \]
Step 4: Integrate both sides \[ \int d(xy) = \int y^2 \, dy \] \[ xy = \frac{y^3}{3} + C \] where \( C \) is an arbitrary constant.
Step 5: Write the general solution \[ \boxed{xy = \frac{y^3}{3} + C} \] or equivalently: \[ xy - \frac{y^3}{3} = C \] Alternative form: \[ 3xy - y^3 = 3C \quad \text{(multiplying both sides by 3)} \] Let \( K = 3C \), then: \[ 3xy - y^3 = K \]