how do you get the solution x^2e^xsiny=c as the solution from (x+2)sinydx=-xcosydy

To solve the given differential equation:

\[
(x+2) \sin y \, dx = -x \cos y \, dy
\]

we can follow these steps:

### Step 1: Separate variables
We aim to separate the variables \(x\) and \(y\) so that each side of the equation contains terms involving only \(x\) or only \(y\).

First, rewrite the equation:

\[
(x+2) \sin y \, dx = -x \cos y \, dy
\]

Divide both sides by \(\sin y\) and by \(x \cos y\) to get:

\[
\frac{(x+2) \, dx}{x} = -\frac{\cos y \, dy}{\sin y}
\]

This simplifies to:

\[
\frac{x+2}{x} \, dx = -\cot y \, dy
\]

### Step 2: Simplify the \(x\)-terms
Break the left-hand side into two fractions:

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

### Step 3: Integrate both sides
Now, integrate both sides separately.

- For the left-hand side:

\[
\int \left( 1 + \frac{2}{x} \right) \, dx = \int 1 \, dx + \int \frac{2}{x} \, dx
\]

This gives:

\[
x + 2 \ln |x|
\]

- For the right-hand side:

\[
\int -\cot y \, dy = \ln |\sin y|
\]

### Step 4: Combine the results
We now have:

\[
x + 2 \ln |x| = \ln |\sin y| + C
\]

Where \(C\) is the constant of integration. 

### Step 5: Exponentiate to simplify
Exponentiate both sides to remove the logarithms:

\[
e^{x + 2 \ln |x|} = e^{\ln |\sin y| + C}
\]

This simplifies to:

\[
e^x |x|^2 = A |\sin y|
\]

where \(A = e^C\) is a new constant. We can write this as:

\[
x^2 e^x = A \sin y
\]

Finally, multiply both sides by a constant factor (absorbing it into \(A\)) to obtain the general solution:

\[
x^2 e^x \sin y = C
\]

where \(C\) is a constant.

### Final solution:
Thus, the solution to the differential equation is:

\[
x^2 e^x \sin y = C
\]

Do you want any more details, or do you have any questions?

Here are 5 questions related to this solution:

1. How do you handle the integration of \(\cot y\) in other contexts?
2. What happens if you change the original equation's coefficients?
3. Can this solution be interpreted geometrically?
4. How does the solution behave as \(x\) becomes very large?
5. What if the initial equation had a different trigonometric function?

**Tip:** Always check if separating variables is feasible before attempting other solution methods.

Explore the step-by-step solution of the differential equation (x+2)sinydx=-xcosydy, which results in the solution x^2e^xsin y = C. This involves separation of variables, logarithmic integration, and trigonometric manipulation, ideal for undergraduate students studying advanced calculus.

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