3.8 Q-14

Question Statement

Solve the differential equation:

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

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Background and Explanation

This is a first-order linear differential equation. To solve it, we will rearrange the terms to separate variables, making it easier to integrate both sides. The key concept here is recognizing that the equation can be simplified by isolating $\frac{dy}{dx}$ and then separating the variables $x$ and $y$ for integration.

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Solution

1. Start with the given equation:

   The equation is:

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

2. Rearrange the terms:

   First, move the terms involving $\frac{dy}{dx}$ to one side:

$$-2 \frac{d y}{d x} - x \frac{d y}{d x} = 2 y^{2} - y$$

3. Factor out $\frac{dy}{dx}$:

   Group the $\frac{dy}{dx}$ terms together:

$$(-2 + x) \frac{d y}{d x} = 2 y^{2} - y$$

4. Separate the variables:

   Divide both sides to separate the variables $x$ and $y$:

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

5. Rewrite the equation:

   We can rewrite the left-hand side as:

$$\left[ \frac{1}{y} + \frac{1}{1 - 2y} \right] , dy = \frac{1}{1 + 2x} , dx$$

6. Integrate both sides:

   Now, we can integrate both sides:

   • On the left-hand side, use the standard integrals:

     • $\int \frac{1}{y} , dy = \ln y$
     • $\int \frac{1}{1 - 2y} , dy = -\frac{1}{2} \ln |1 - 2y|$

     Thus, the left-hand side becomes:

$$\ln y - \frac{1}{2} \ln |1 - 2y|$$

• On the right-hand side, integrate:

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

Therefore, we have:

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

7. Simplify the equation:

   Combine the logarithms:

$$\ln \left( \frac{y}{1 - 2y} \right) = \ln C (x + 2)$$

Exponentiate both sides to remove the logarithms:

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

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Key Formulas or Methods Used

• Separation of Variables: Rearranged the equation so that all terms involving $x$ were on one side and all terms involving $y$ were on the other.
• Integration of Rational Functions: Used basic integration techniques for functions involving $y$ and $x$.
• Logarithmic Properties: Used properties of logarithms to combine and simplify the equation.

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Summary of Steps

1. Start with the equation: $y - x \frac{d y}{d x} = 2 \left( y^{2} + \frac{d y}{d x} \right)$.
2. Rearrange to: $-2 \frac{d y}{d x} - x \frac{d y}{d x} = 2 y^{2} - y$.
3. Factor out $\frac{dy}{dx}$: $(-2 + x) \frac{d y}{d x} = 2 y^{2} - y$.
4. Separate the variables: $\frac{1}{y(1 - 2y)} , dy = \frac{1}{1 + 2x} , dx$.
5. Rewrite as: $\left[ \frac{1}{y} + \frac{1}{1 - 2y} \right] , dy = \frac{1}{1 + 2x} , dx$.
6. Integrate both sides: $\ln y - \frac{1}{2} \ln |1 - 2y| = \frac{1}{2} \ln |1 + 2x| + \ln C$.
7. Simplify to: $\frac{y}{1 - 2y} = C(x + 2)$.