Question

\(\mathrm{y}^{2}=(\mathrm{x}-\mathrm{c})^{3}\) is general solution of the differential equation: (where c is arbitrary constant). (A) \((\mathrm{dy} / \mathrm{dx})^{3}=27 \mathrm{y}\) (B) \(2(\mathrm{dy} / \mathrm{dx})^{3}-8 \mathrm{y}=0\) (C) \(8(\mathrm{dy} / \mathrm{dx})^{3}=27 \mathrm{y}\) (D) \(8\left(\mathrm{~d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)-27 \mathrm{y}=0\)

Short answer

In conclusion, none of the given options match the derived equation for the given implicit function.

Step-by-step solution

Differentiate the given function implicitly with respect to x

Start by differentiating the given function with respect to x using the chain rule and implicit differentiation. Given function: \(y^2 = (x-c)^3\) Differentiate implicitly with respect to x: \(2y\frac{dy}{dx} = 3(x-c)^2 \frac{d(x-c)}{dx}\) Since \(\frac{d(x-c)}{dx} = 1\), the equation can be simplified as follows: \(2y\frac{dy}{dx} = 3(x-c)^2\)

Solve for \(\frac{dy}{dx}\)

Now, isolate \(\frac{dy}{dx}\) to make comparing with the given options easier: \(\frac{dy}{dx} = \frac{3(x-c)^2}{2y}\)

Substitute the expression for y from the original equation

Substitute the expression for y from the original equation into the above equation to eliminate y: Recall the given function: \(y^2 = (x-c)^3\) So, \(y = \sqrt{(x-c)^3}\) Now, substitute this expression for y into the equation we derived: \(\frac{dy}{dx} = \frac{3(x-c)^2}{2\sqrt{(x-c)^3}}\)

Simplify the equation further

Further simplify the equation to get a simple expression for \(\frac{dy}{dx}\): \(\frac{dy}{dx} = \frac{3}{2} \cdot{(x-c)^{\frac{2-3}{2}}}\) \(\frac{dy}{dx} = \frac{3}{2} \cdot{(x-c)^{-\frac{1}{2}}}\)

Examine the given options

Now, look at the given options and compare with the derived equation for \(\frac{dy}{dx}\). In this case, none of the provided choices match our derived equation. Therefore, at this point, we must conclude that there is not an option that corresponds to our derived equation. In conclusion, none of the given options match the derived equation for the given implicit function.

Key concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables cannot be easily separated. Sometimes, an equation is not given in an explicit form like \[y = f(x)\]Instead, you may encounter implicit forms like \[y^2 = (x-c)^3\]In such cases, implicit differentiation comes in handy. Here's what you do:

• Differentiate each side of the equation with respect to the independent variable, here it's \(x\).
• Apply derivative rules, like the chain rule, to handle expressions with \(y\).
• Remember to multiply by \(\frac{dy}{dx}\) whenever you differentiate any terms involving \(y\).

Implicit differentiation allows us to work through complex-equation differentiation smoothly, even when \(y\) isn't alone.

Chain Rule

The chain rule is a fundamental rule in calculus used to find the derivative of composite functions. When you have a function nested within another function, the chain rule helps you differentiate it. Consider how it applies here:Given \[y^2 = (x-c)^3\]When differentiating with respect to \(x\), if you have a term like \(y^2\), apply the chain rule:

• Differentiating \(y^2\) gives \(2y\cdot\frac{dy}{dx}\).
• The chain comes into play as you differentiate the outer function \((y^2)\) and multiply it by the derivative of the inner function \(y\), giving rise to \(\frac{dy}{dx}\).

Similarly, when differentiating \((x-c)^3\):

• First, differentiate the outer function resulting in \(3(x-c)^2\).
• Since \((x-c)\) is the inner function, its derivative \(\frac{d(x-c)}{dx}\) is just 1, simplifying your work here.

The chain rule is crucial for differentiating nested or composite functions efficiently.

Solution of Differential Equations

When solving differential equations, the aim is to find a function that satisfies the equation. In the original problem:\[y^2 = (x-c)^3\]Differentiating and manipulating this expression lets us find derivatives like \(\frac{dy}{dx}\) in terms of \(x\). The process involves:

• Implicit differentiation to handle \(y^2\).
• Using algebraic manipulations to isolate derivatives.
• Substituting back expressions from the original equation to simplify terms.

These steps elevate your understanding and help reach an expression compatible (or not) with options given in multiple-choice settings. If none match, ensure reevaluation of potential typographical errors, or interpretation issues, as was found in our example.

Arbitrary Constant

In differential equations, an arbitrary constant is typically a constant term that appears as part of the general solution. It represents an unknown but fixed quantity. For example, in our exercise, \(c\) acts as such a constant.Consider:\[y^2 = (x-c)^3\]The constant \(c\):

• Reflects the property of integration that solutions are not unique but part of a family varying by constant shifts.
• Acts as a placeholder indicating this family of solutions, adjustable based on initial conditions or boundary values.

Having a grasp of arbitrary constants helps understand why solutions may take different forms yet remain valid under broader circumstances, tailored by context or additional specifications.