The equations given require elimination of the arbitrary constants present in them. Here's the breakdown for each:

1. $2y^2 = 16bx$:

   • To eliminate the constant $b$, you could express it in terms of $x$ and $y$. From $2y^2 = 16bx$, you can solve for $b$ as follows: $b = \frac{y^2}{8x}$ However, this approach doesn't eliminate the constant completely as it's just expressed differently. Typically, without additional conditions or information, constants in such an equation remain as parameters.

2. $y = 21 + Ae^{-2x}$:

   • Here $A$ is a constant that you could solve for if a specific $x, y$ pair was given. Otherwise, the constant remains part of the general solution.

3. $y = \sin(x + b)$:

   • To eliminate $b$, you would typically need boundary conditions such as specific values for $x$ and $y$ (e.g., $y = \sin(x_0 + b) = k$ for known $x_0$ and $k$). Without additional information, $b$ remains an arbitrary phase shift in the sine function.

4. $3x^2y - 2x = cy$:

   • Solving for $c$ would again need a known pair $(x, y)$ to plug in and find $c$ explicitly. Without it, $c$ just indicates the dependency between $x, y$ and cannot be eliminated algebraically from this form.

5. $y = c_1e^{3x} + c_2e^{3x}$:

   • Here, $c_1$ and $c_2$ are constants. We can simplify this by letting $c = c_1 + c_2$ since the exponential terms are the same: $y = ce^{3x}$ Where $c$ is the new constant which is the sum of $c_1$ and $c_2$.

Eliminating arbitrary constants typically requires specific conditions or additional information like boundary or initial values. If the context or problem statement specifies such conditions, those can be used to determine the constants.

Would you like to delve into any particular equation further, or do you have more questions on this topic?

Related questions to consider:

1. How would additional boundary conditions affect the elimination of constants in these equations?
2. What are practical situations where you might encounter equations like these, and how would you handle them?
3. Can you explore solutions to these equations graphically?
4. How does the presence of exponential terms in equation 2 and 5 affect the behavior of the solutions?
5. What techniques are generally used in calculus to deal with arbitrary constants in differential equations?

Tip:

When trying to eliminate constants, consider the physical or practical context of the problem which might suggest specific values or relationships.