Question

Find a first order differential equation for the given family of curves. $$ y=x^{3}+\frac{c}{x} $$

Short answer

Question: Find the first order differential equation for the family of curves given by the equation $$y = x^3 + \frac{c}{x}$$, where c is a constant. Solution: The first order differential equation for the given family of curves is $$x\frac{dy}{dx} = 4x^3 - y$$.

Step-by-step solution

Differentiate the given equation with respect to x.

To find the first order differential equation, we'll start by differentiating both sides of the equation with respect to x. Using the power rule and the quotient rule for differentiation, we get: $$ \frac{dy}{dx} = \frac{d}{dx}(x^3) + \frac{d}{dx}\left(\frac{c}{x}\right) $$

Perform the differentiation and simplify the expression.

Now we'll perform the differentiation on each term: $$ \frac{dy}{dx} = 3x^2 - \frac{c}{x^2} $$ We have obtained the first derivative of the given equation. The next step is to eliminate the constant c from the expression.

Eliminate the constant c from the expression.

To eliminate c, we'll express it in terms of y and x using the given equation: $$ c = x(y - x^3) $$ Now, we'll substitute this expression for c into the expression we obtained in Step 2: $$ \frac{dy}{dx} = 3x^2 - \frac{x(y - x^3)}{x^2} $$

Simplify the expression and write the final first order differential equation.

Now we'll simplify the expression by canceling out the x terms and combining like terms: $$ \frac{dy}{dx} = 3x^2-\frac{(y-x^3)}{x} $$ Multiplying both sides by x, we get: $$ x\frac{dy}{dx} = 3x^3 - (y - x^3) $$ Thus, the first order differential equation for the given family of curves is: $$ x\frac{dy}{dx} = 4x^3 - y $$

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with the rate of change of a quantity. In simple terms, it is the process of finding the derivative of a function. When we differentiate a function, we are finding how the function's output changes with respect to one of its inputs.
In the context of our original exercise, differentiation helps us find the rate of change of the curve described by the equation \(y = x^3 + \frac{c}{x}\). By taking the derivative with respect to \(x\), we aim to simplify and express the relation between \(y\) and \(x\) without the constant \(c\).

• Differentiation allows us to find the slope of a curve at any given point.
• It is crucial for understanding and generating differential equations, which express relations between functions and their derivatives.
• In solving the exercise, differentiation was the first step toward achieving a first order differential equation.

Power Rule

The power rule is a simple yet powerful technique used in differentiation. It is used for differentiating expressions where a variable is raised to a constant power. The general form of the power rule states that if you have a function \(f(x) = x^n\), then its derivative \(f'(x) = nx^{n-1}\).
In the original exercise, the power rule was used to find the derivative of \(x^3\), where applying the rule gives us \(3x^2\) because \(n=3\).
Here's a quick breakdown of the power rule in application:

• Identify the power of \(x\) in the function you need to differentiate.
• Multiply the entire term by the power of \(x\).
• Decrease the exponent of \(x\) by 1.

Using the power rule simplifies the process of differentiation, making it a fast and efficient way to handle polynomial expressions.

Quotient Rule

The quotient rule is used for differentiating functions that are the ratio of two differentiable functions. It's particularly useful when the function you want to differentiate is in the form \(\frac{u}{v}\). The formula for the quotient rule is given by \(\left(\frac{u}{v}\right)' = \frac{vu'-uv'}{v^2}\), where \(u\) and \(v\) are functions of \(x\).
In the exercise, the quotient rule was applied to differentiate the term \(\frac{c}{x}\), where \(c\) is a constant considered as the function \(u(x)\) and \(x\) is the function \(v(x)\). Applying the quotient rule gave us \(-\frac{c}{x^2}\), which was crucial for obtaining the first order differential equation.

• Ensure both numerator and denominator functions are differentiable.
• Compute separately the derivatives of the numerator and denominator functions.
• Substitute and simplify using the formula.

While the quotient rule may look complex, it becomes much simpler with practice and helps tackle a wide variety of division expressions efficiently.

Simplifying Differential Equations

Simplifying differential equations involves reducing them to a form that is easier to interpret or solve. In our problem, this meant expressing the equation without the arbitrary constant \(c\) and manipulating terms to reveal the underlying relationship between \(x\) and \(y\).
After differentiation, we had \(\frac{dy}{dx} = 3x^2 - \frac{c}{x^2}\). Then, by eliminating \(c\), we express \(c\) as \(x(y - x^3)\) and substitute it back, which eventually leads us to the simpler form: \(x\frac{dy}{dx} = 4x^3 - y\).
This simplification helps:

• Highlight the relationship between the variables involved.
• Make the equation more practical for solving or further analysis.
• Facilitate the integration or numerical techniques if used in practical applications.

Thus, simplifying the differential equation not only makes it manageable, but also reveals more about the dynamics it represents in a clearer way.