Find `dy/dx if 2x-3y=e^x`

Answer

To find \(\frac{dy}{dx}\) for the equation \(2x - 3y = e^x\), we will differentiate both sides of the equation with respect to \(x\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ 2x - 3y = e^x \]

2. **Differentiate both sides with respect to \(x\):**
- The derivative of \(2x\) is \(2\).
- The derivative of \(-3y\) is \(-3\frac{dy}{dx}\) (using the chain rule).
- The derivative of \(e^x\) is \(e^x\).

So, differentiating both sides gives:
\[ \frac{d}{dx}(2x) - \frac{d}{dx}(3y) = \frac{d}{dx}(e^x) \]
This results in:
\[ 2 - 3\frac{dy}{dx} = e^x \]

3. **Rearrange the equation to isolate \(\frac{dy}{dx}\):**
\[ -3\frac{dy}{dx} = e^x - 2 \]

4. **Solve for \(\frac{dy}{dx}\):**
Divide both sides by \(-3\):
\[ \frac{dy}{dx} = \frac{2 - e^x}{3} \]

### Final Result:
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{2 - e^x}{3} \]