3.8 Q-8

Question Statement

Solve the differential equation:

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

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Background and Explanation

This is a first-order differential equation where we can solve for $y$ in terms of $x$ using the method of separation of variables. The goal is to separate the terms involving $y$ on one side and the terms involving $x$ on the other side, then integrate both sides.

We will also need to apply logarithmic properties and integrate simple rational functions during the process.

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Solution

1. Start with the equation:

   The given equation is:

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

2. Separate the variables:

   To separate the variables, first multiply both sides of the equation by $y$:

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

Now, rearrange to get all $y$-terms on the left and $x$-terms on the right:

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

Then, distribute the terms:

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

3. Integrate both sides:

   Now, integrate both sides of the equation. On the left side:

$$\int \left( x + \frac{1}{x} \right) dx = \frac{x^2}{2} + \ln |x|$$

On the right side:

$$\int \left( 1 + \frac{1}{y} \right) dy = y + \ln |y|$$

So, we now have:

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

4. Simplify the equation:

   Combine the logarithmic terms on the right-hand side:

$$y + \ln |y| = \frac{x^2}{2} + \ln (Cx)$$

Exponentiate both sides to remove the logarithms:

$$\frac{y e^y}{Cx} = e^{\frac{x^2}{2}}$$

Finally, rearrange to get the solution:

$$y e^y = Cx e^{\frac{x^2}{2}}$$

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Key Formulas or Methods Used

• Separation of Variables: Rearranged the terms to separate the variables $y$ and $x$.
• Integration: Integrated terms like $\frac{1}{y}$ and $\frac{1}{x}$, and applied the standard integrals for these functions.
• Logarithmic Properties: Used properties of logarithms to combine constants and simplify the equation.
• Exponentiation: Exponentiated both sides to eliminate the natural logarithms.

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Summary of Steps

1. Start with the equation: $\frac{x^2 + 1}{y + 1} = \frac{x}{y} \frac{dy}{dx}$.
2. Separate variables: $\left( x + \frac{1}{x} \right) dx = \left( x + \frac{1}{y} \right) dy$.
3. Integrate both sides:
   • Left side: $\frac{x^2}{2} + \ln |x|$.
   • Right side: $y + \ln |y|$.
4. Combine logarithmic terms: $y + \ln |y| = \frac{x^2}{2} + \ln (Cx)$.
5. Exponentiate and solve: $y e^y = Cx e^{\frac{x^2}{2}}$.