3.8 Q-10

Question Statement

Solve the differential equation:

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

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Background and Explanation

This is a first-order separable differential equation. To solve it, we need to separate the variables $x$ and $y$, integrate both sides, and then simplify the result. We will use standard integration formulas for polynomials and rational functions during the solution process.

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Solution

1. Start with the given equation:

   The equation is:

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

2. Separate the variables:

   To separate the variables, we divide both sides by $2 x^2 y$:

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

Now multiply both sides by $2y$ to isolate $dy$ and $dx$ on opposite sides:

$$2y , dy = \frac{x^2 - 1}{x^2} , dx$$

3. Integrate both sides:

   On the left side, we have the integral of $2y$, which is straightforward:

$$\int 2y , dy = y^2$$

On the right side, split the fraction $\frac{x^2 - 1}{x^2}$ into two parts:

$$\frac{x^2 - 1}{x^2} = 1 - \frac{1}{x^2}$$

So, the right-hand side becomes:

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

Now, integrate both terms:

• The integral of $1$ is $x$.
• The integral of $\frac{1}{x^2}$ is $-\frac{1}{x}$.

So, the right-hand side becomes:

$$x - \frac{1}{x}$$

4. Combine results:

   Putting both sides together:

$$y^2 = x - \frac{1}{x} + C$$

where $C$ is the constant of integration.

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Key Formulas or Methods Used

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

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Summary of Steps

1. Start with the equation: $2x^2 y \frac{dy}{dx} = x^2 - 1$.
2. Separate the variables: $2y , dy = \frac{x^2 - 1}{x^2} , dx$.
3. Integrate both sides:
   • Left side: $\int 2y , dy = y^2$.
   • Right side: $\int 1 , dx - \int \frac{1}{x^2} , dx = x - \frac{1}{x}$.
4. Final result: $y^2 = x - \frac{1}{x} + C$.