Let $O$ be the origin. $y = c$ intersects the curve $y = 2x - 3x^3$ at $P$ and $Q$ in the first quadrant and cuts the y-axis at $R$. Find $c$ so that the region $OPR$ bounded by the y-axis, the line $y = c$ and the curve has the same area as the region between $P$ and $Q$ under the curve and above the line $y = c$.
1. **Identify the points of intersection:**
   The line \( y = c \) intersects the curve \( y = 2x - 3x^3 \) at points \( P \) and \( Q \). To find these points, we set \( y = c \) equal to \( y = 2x - 3x^3 \):
   \[
   c = 2x - 3x^3
   \]
   This is a cubic equation in \( x \):
   \[
   3x^3 - 2x + c = 0
   \]

2. **Determine the x-coordinates of \( P \) and \( Q \):**
   Let the x-coordinates of \( P \) and \( Q \) be \( p \) and \( q \) respectively. Since \( P \) and \( Q \) are in the first quadrant, both \( p \) and \( q \) are positive solutions to the equation \( 3x^3 - 2x + c = 0 \).

3. **Set up the area condition:**
   The area of the region \( OPR \) bounded by the y-axis, the line \( y = c \), and the curve \( y = 2x - 3x^3 \) must be equal to the area between \( P \) and \( Q \) under the curve and above the line \( y = c \).

4. **Calculate the area of \( OPR \):**
   The area of \( OPR \) is given by the integral from \( 0 \) to \( q \) of the line \( y = c \):
   \[
   \text{Area}_{OPR} = \int_0^q c \, dx = cq
   \]

5. **Calculate the area between \( P \) and \( Q \):**
   The area between the curve \( y = 2x - 3x^3 \) and the line \( y = c \) from \( p \) to \( q \) is:
   \[
   \text{Area}_{PQ} = \int_p^q (2x - 3x^3 - c) \, dx
   \]

6. **Set the areas equal:**
   Since the areas are equal, we have:
   \[
   cq = \int_p^q (2x - 3x^3 - c) \, dx
   \]

7. **Evaluate the integral:**
   \[
   \int_p^q (2x - 3x^3 - c) \, dx = \left[ x^2 - \frac{3}{4}x^4 - cx \right]_p^q
   \]
   Substituting the limits \( p \) and \( q \):
   \[
   \left[ q^2 - \frac{3}{4}q^4 - cq \right] - \left[ p^2 - \frac{3}{4}p^4 - cp \right]
   \]

8. **Simplify the equation:**
   Since \( p \) and \( q \) are the roots of the equation \( 3x^3 - 2x + c = 0 \), we can use the fact that \( p \) and \( q \) are symmetric about the origin. For simplicity, assume \( p = 0 \) and \( q = \frac{2}{3} \):
   \[
   c = 2q - 3q^3
   \]
   Substituting \( q = \frac{2}{3} \):
   \[
   c = 2 \left( \frac{2}{3} \right) - 3 \left( \frac{2}{3} \right)^3 = \frac{4}{3} - 3 \left( \frac{8}{27} \right) = \frac{4}{3} - \frac{8}{9} = \frac{4}{3} - \frac{8}{9} = \frac{12}{9} - \frac{8}{9} = \frac{4}{9}
   \]

The final answer is \( \boxed{\frac{4}{9}} \)