3.8 Q-9

Question Statement

Solve the differential equation:

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

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Background and Explanation

This is a first-order separable differential equation, which means we can separate the variables $y$ and $x$ to integrate both sides. In this case, we’ll need to rearrange the terms and use standard integration techniques for both $y$-terms and $x$-terms. The integral of $\frac{1}{1 + y^2}$ is a standard arctangent formula.

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Solution

1. Start with the given equation:

   The original equation is:

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

2. Separate the variables:

   Multiply both sides of the equation by $x$ to get the $x$-terms and $y$-terms on opposite sides:

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

3. Integrate both sides:

   Now, integrate both sides of the equation. On the left side, the integral of $\frac{1}{1 + y^2}$ is $\tan^{-1}(y)$, a standard integral:

$$\int \frac{1}{1 + y^2} , dy = \tan^{-1}(y)$$

On the right side, the integral of $\frac{x}{2}$ is $\frac{x^2}{4}$:

$$\int \frac{x}{2} , dx = \frac{x^2}{4}$$

So we have:

$$\tan^{-1}(y) = \frac{x^2}{4} + C_1$$

where $C_1$ is the constant of integration.

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Key Formulas or Methods Used

• Separation of Variables: Rearranged the terms so that all $y$-terms are on one side and all $x$-terms are on the other side.
• Integration: Used standard integrals for $\frac{1}{1 + y^2}$ and $x$.
  • $\int \frac{1}{1 + y^2} , dy = \tan^{-1}(y)$
  • $\int x , dx = \frac{x^2}{2}$
• Arctangent Identity: Recognized that the integral of $\frac{1}{1 + y^2}$ is $\tan^{-1}(y)$.

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Summary of Steps

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