3.8 Q-4

Question Statement

Solve the following differential equation:

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

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Background and Explanation

This is a first-order differential equation that can be solved using the method of separation of variables. In this method, we aim to separate the terms involving $x$ and $y$ on opposite sides of the equation so that we can integrate both sides.

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Solution

1. Rearrange the equation:

   Start with the given equation:

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

To separate the variables, multiply both sides by $y , dx$:

$$y , dy = (1 - x) , dx$$

2. Integrate both sides:

   Now, integrate both sides:

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

The left-hand side integrates to $\frac{y^2}{2}$, and the right-hand side integrates as follows:

$$\int 1 , dx = x \quad \text{and} \quad \int x , dx = \frac{x^2}{2}$$

So, we get:

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

3. Simplify the equation:

   Multiply both sides by 2 to simplify:

$$y^2 = 2x - x^2 + 2C_1$$

4. Final equation:

   Finally, rewrite the constant term as $C$ (since $2C_1$ is just another constant):

$$y^2 = x(2 - x) + C$$

This is the general solution to the differential equation.

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Key Formulas or Methods Used

• Separation of variables: Rearranged the equation to separate the variables $x$ and $y$.
• Integration: Integrated both sides with respect to their respective variables.
• Constant of integration: Used the constant $C_1$ (which we later simplified to $C$) to account for the indefinite integral.

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Summary of Steps

1. Rearrange the equation: $y , dy = (1 - x) , dx$
2. Integrate both sides: $\frac{y^2}{2} = x - \frac{x^2}{2} + C_1$
3. Simplify the equation: $y^2 = 2x - x^2 + 2C_1$
4. Final equation: $y^2 = x(2 - x) + C$