3.8 Q-12

Question Statement

Solve the differential equation:

$$\left(x^{2} - x^{2}y\right) \frac{dy}{dx} + y^{2} + xy^{2} = 0$$

---

Background and Explanation

This is a first-order separable differential equation. The key to solving it is isolating the terms involving $y$ on one side and those involving $x$ on the other side. Once separated, we integrate both sides to find the solution.

---

Solution

1. Start with the given equation:

   The equation is:

$$x^{2}(1 - y) \frac{dy}{dx} + y^{2}(1 + x) = 0$$

2. Rearrange the equation:

   Move $y^{2}(1 + x)$ to the other side:

$$x^{2}(1 - y) \frac{dy}{dx} = -y^{2}(1 + x)$$

3. Separate the variables:

   To separate the variables, divide both sides by $x^{2}(1 - y) y^{2}$:

$$\left( \frac{1 - y}{-y^{2}} \right) dy = \left( \frac{1 + x}{x^{2}} \right) dx$$

4. Simplify the expression:

   Simplifying further:

$$\left( \frac{1}{y} - \frac{1}{y^{2}} \right) dy = \left( \frac{1}{x^{2}} - \frac{1}{x} \right) dx$$

5. Integrate both sides:

   Now, integrate both sides:

$$\int \left( \frac{1}{y} - \frac{1}{y^{2}} \right) dy = \int \left( \frac{1}{x^{2}} - \frac{1}{x} \right) dx$$

On the left-hand side:

$$\int \frac{1}{y} dy - \int \frac{1}{y^{2}} dy = \ln|y| + \frac{1}{y}$$

On the right-hand side:

$$\int \frac{1}{x^{2}} dx - \int \frac{1}{x} dx = -\frac{1}{x} + \ln|x|$$

6. Combine the results:

   Putting both sides together:

$$\ln|y| + \frac{1}{y} = -\frac{1}{x} + \ln|x| + C$$

Where $C$ is the constant of integration.

7. Final simplified equation:

   The final solution can be written as:

$$\ln|x| - \frac{1}{x} + C$$

---

Key Formulas or Methods Used

• Separation of Variables: Rearranged the equation to separate $y$-terms and $x$-terms.
• Integration:
  • $\int \frac{1}{y} , dy = \ln|y|$
  • $\int \frac{1}{y^2} , dy = -\frac{1}{y}$
  • $\int \frac{1}{x^2} , dx = -\frac{1}{x}$
  • $\int \frac{1}{x} , dx = \ln|x|$

---

Summary of Steps

1. Start with the equation: $x^{2}(1 - y) \frac{dy}{dx} + y^{2}(1 + x) = 0$.
2. Rearrange to: $x^{2}(1 - y) \frac{dy}{dx} = -y^{2}(1 + x)$.
3. Separate the variables: $\left( \frac{1}{y} - \frac{1}{y^{2}} \right) dy = \left( \frac{1}{x^{2}} - \frac{1}{x} \right) dx$.
4. Integrate both sides:
   • Left side: $\ln|y| + \frac{1}{y}$
   • Right side: $-\frac{1}{x} + \ln|x|$
5. Final result: $\ln|y| + \frac{1}{y} = -\frac{1}{x} + \ln|x| + C$.