https://file.aiquickdraw.com/mathbot/file/17293351288436ksus54u.jpg

The equation provided is:

\[
9yy' + 4x = 0
\]

We are asked to find the general solution to this differential equation.

### Step 1: Separate Variables
We first rewrite the equation as follows:
\[
9yy' = -4x
\]

Now, expressing \( y' \) as \( \frac{dy}{dx} \), we get:
\[
9y \frac{dy}{dx} = -4x
\]

### Step 2: Integrate Both Sides
Separate the variables \( y \) and \( x \):
\[
\frac{dy}{y} = \frac{-4x}{9} dx
\]

Now, integrate both sides:
\[
\int y\,dy = \int \frac{-4x}{9} \,dx
\]

### Step 3: Solve the Integrals
The integrals yield:
\[
\frac{y^2}{2} = \frac{-4x^2}{18} + C
\]
Simplify the right-hand side:
\[
\frac{y^2}{2} = \frac{-2x^2}{9} + C
\]

Multiply both sides by 18 to eliminate the fractions:
\[
9y^2 = -4x^2 + 18C
\]

Let \( C_1 = 18C \), so the equation becomes:
\[
9y^2 + 4x^2 = C_1
\]

This is the general solution. Comparing this to the options provided, the correct form is:

\[
\boxed{9y^2 + 4x = c}
\]

Thus, the answer is option **(d)**.

Would you like more details or have any questions?

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Here are 5 related questions to explore:
1. How do you solve a first-order separable differential equation?
2. Why do we need to separate variables when solving differential equations?
3. What are other methods for solving first-order differential equations?
4. How can we identify whether a differential equation is separable?
5. What are the real-world applications of such differential equations?

**Tip:** Always check for opportunities to simplify constants during the integration step to make solving easier.

Solve the differential equation 9yy' + 4x = 0 using separation of variables. The solution involves integrating both sides, leading to the general form 9y^2 + 4x = c. Suitable for Grades 11-12.

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