3.8 Q-11

Question Statement

Solve the differential equation:

$$\frac{dy}{dx} + \frac{2xy}{2y + 1} = x$$

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Background and Explanation

This is a first-order linear differential equation. To solve it, we first isolate $\frac{dy}{dx}$, then separate the variables $x$ and $y$, and finally integrate both sides. Understanding the process of isolating terms and separating variables is key to solving this type of equation.

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Solution

1. Start with the given equation:

   The equation is:

$$\frac{dy}{dx} + \frac{2xy}{2y + 1} = x$$

2. Isolate $\frac{dy}{dx}$:

   To isolate $\frac{dy}{dx}$, subtract $\frac{2xy}{2y + 1}$ from both sides:

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

3. Simplify the expression:

   Combine the terms on the right-hand side:

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

The $2xy$ terms cancel out, leaving:

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

4. Separate the variables:

   Now separate the variables by multiplying both sides by $(2y + 1)$ and $dx$:

$$(2y + 1) , dy = x , dx$$

5. Integrate both sides:

   On the left-hand side, integrate $(2y + 1) , dy$:

$$\int (2y + 1) , dy = \int x , dx$$

The integral of $2y + 1$ with respect to $y$ is:

$$y^2 + y$$

The integral of $x$ with respect to $x$ is:

$$\frac{x^2}{2}$$

6. Combine the results:

   Putting both sides together, we get:

$$y^2 + y = \frac{x^2}{2} + C_1$$

Where $C_1$ is the constant of integration.

7. Rearrange the equation:

   Finally, we can rewrite the equation as:

$$y(y + 1) = \frac{x^2}{2} + C_1$$

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Key Formulas or Methods Used

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

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Summary of Steps

1. Start with the equation: $\frac{dy}{dx} + \frac{2xy}{2y + 1} = x$.
2. Isolate $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{x}{2y + 1}$.
3. Separate the variables: $(2y + 1) , dy = x , dx$.
4. Integrate both sides:
   • Left side: $\int (2y + 1) , dy = y^2 + y$
   • Right side: $\int x , dx = \frac{x^2}{2}$
5. Final result: $y(y + 1) = \frac{x^2}{2} + C_1$.