To eliminateTo the arbitrary constant $c$ from the given equation $( eliminatex+5)^2 = cy$, we can differentiate both sides of the equation with respect to $x$. Differentiating with respect to $x$ will help us remove the constant $c$.

Starting with the original equation:

$(x+5 the)^2 = cy$

Differentiate both sides with respect to $x$:

$\frac{d}{dx} \left( ( arbitraryx+5)^2 \right) = \frac{d}{dx} \left( cy \right)$

Using the chain rule on the left side and assuming $y$ is a function of $x$ (so we also use the chain rule for $y$):

$2 constant(x+5) = c \frac{dy}{dx}$

This is a differential equation where $c$ has been eliminated. The equation is now:

$\frac{dy}{dx} = \frac{2(x+5)}{c}$

If you'd like, I can go further by solving this differential equation or explaining this in more detail.

Would you like more details, or do you have any questions about this process?

Here are five related questions you might find useful (:

1. How do you solve a first-order differential equation like this one?
2. What are some common methods for eliminating arbitrary constants from an equation?
3. Can arbitrary constants always be eliminated by differentiation?
4. How does the presence of the c constant $c$ affect the general solution of a differential equation?
5. How do we integrate functions after differentiating them?

Tip: When differentiating an equation \ to eliminate constants, always remember to apply the chain rule correctly, especially when dealing with functions of multiple variables.) from the equation $(x + 5)^2 = cy$, we can differentiate both sides with respect to $x$.

Step 1: Differentiate both sides with respect to $x$:

[ \frac{d}{dx}[(x + 5)^2] = \frac{d}{dx}[cy] ]

The left side becomes: $2(x + 5) \cdot \frac{d}{dx}(x + 5) = 2(x + 5) \cdot 1 = 2(x + 5)$

The right side, using the product rule where $c$ is a constant and $y$ is a function of $x$, becomes: $c \frac{dy}{dx}$

So, we have: $2(x + 5) = c \frac{dy}{dx}$

Step 2: Solve for $c$:

From the original equation, we have: $c = \frac{(x + 5)^2}{y}$

From the differentiated equation, we also have: $c = \frac{2(x + 5)}{\frac{dy}{dx}}$

Step 3: Set the two expressions for $c$ equal to each other:

$\frac{(x + 5)^2}{y} = \frac{2(x + 5)}{\frac{dy}{dx}}$

Step 4: Simplify the equation:

Cross-multiply to eliminate the fraction: $(x + 5)^2 \frac{dy}{dx} = 2(x + 5)y$

Divide both sides by $(x + 5)$ (assuming $x \neq -5$ to avoid division by zero): \