Certainly! Let's solve the differential equation step by step:

Problem Statement

Solve the differential equation: $\cos(x) \, dx + \left(1 + \frac{2}{y}\right) \sin(x) \, dy = 0$

Solution

1. Separate Variables:
   We want to separate the variables $x$ and $y$. Start by dividing both sides by $\sin(x)$ (assuming $\sin(x) \neq 0$): $\frac{\cos(x)}{\sin(x)} \, dx + \left(1 + \frac{2}{y}\right) \, dy = 0$

   Simplify the equation: $\cot(x) \, dx + \left(1 + \frac{2}{y}\right) \, dy = 0$

2. Rearrange Terms: To separate variables, move the terms involving $y$ to one side and the terms involving $x$ to the other: $\cot(x) \, dx = -\left(1 + \frac{2}{y}\right) \, dy$

3. Integrate Both Sides:
   Now, integrate both sides of the equation separately.

   • For the left side: $\int \cot(x) \, dx = \int \frac{\cos(x)}{\sin(x)} \, dx$ The integral of $\cot(x)$ is $\ln|\sin(x)|$: $\ln|\sin(x)| = \text{C}_1$

   • For the right side: $\int -\left(1 + \frac{2}{y}\right) \, dy = -\int 1 \, dy - \int \frac{2}{y} \, dy$ The integrals are: $-y - 2\ln|y| = \text{C}_2$

4. Combine the Results:
   The equation after integration becomes: $\ln|\sin(x)| = -y - 2\ln|y| + C$ where $C = \text{C}_1 + \text{C}_2$ is the constant of integration.

5. Simplify the Expression:
   You can express the solution more compactly: $\ln|\sin(x)| + y + 2\ln|y| = C$

   Alternatively, exponentiate both sides to eliminate the logarithms, if needed: $|\sin(x)| \cdot y^2 \cdot e^y = e^C$ This can be simplified as: $|\sin(x)| \cdot y^2 \cdot e^y = K$ where $K = e^C$ is another constant.

Final Answer

The general solution to the differential equation is: $|\sin(x)| \cdot y^2 \cdot e^y = K$ or equivalently: $\ln|\sin(x)| + y + 2\ln|y| = C$ where $C$ and $K$ are constants of integration.

Writing Tip

When writing this solution on paper:

• Clearly separate each step.
• Show the integration process explicitly, including the constants of integration.
• Make sure to explain any assumptions, like $\sin(x) \neq 0$.
• Write clearly and ensure all logarithmic and exponential steps are clearly stated.
• Box or highlight the final answer to make it stand out.

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Additional Questions to Consider:

1. What if $\sin(x) = 0$? How does it affect the solution?
2. How would the solution change if the differential equation had different initial conditions?
3. What are the implications of the constant $C$ in the general solution?
4. Can this method be applied to similar differential equations with different functions?
5. How would you solve the equation if $\cos(x)$ and $\sin(x)$ were replaced by other trigonometric functions?

Tip:

Always double-check each integration step to ensure constants and coefficients are correctly handled.