Question

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

Short answer

Question: Find the first order differential equation that represents the family of curves described by the equation $$e^{xy} = cy$$. Answer: The first order differential equation that represents the given family of curves is $$\frac{dy}{dx} = \frac{-y}{x - \frac{1}{x}}$$.

Step-by-step solution

Calculate the derivative of the given equation with respect to x

To find the first order differential equation, we will take the derivative of the given equation with respect to $$x$$. We will employ the chain rule and the product rule for differentiation: $$ \frac{d}{dx}(e^{xy}) = \frac{d}{dx}(cy) $$ Using the chain rule: $$ (e^{xy})(y + x\frac{dy}{dx}) = c\frac{dy}{dx} $$

Solve for dy/dx

Now, we need to solve for $$\frac{dy}{dx}$$. First, let's rewrite the equation by replacing $$e^{xy}$$ with $$cy$$ (from the initial equation). $$ (cy)(y + x\frac{dy}{dx}) = c\frac{dy}{dx} $$ Now, to isolate the $$\frac{dy}{dx}$$ term, we can divide both sides by $$cx$$: $$ y + x\frac{dy}{dx} = \frac{\frac{dy}{dx}}{x} $$ Move the $$\frac{dy}{dx}$$ term to the left-hand side and the $$y$$ term to the right-hand side: $$ x\frac{dy}{dx} - \frac{\frac{dy}{dx}}{x} = -y $$ Take the common factor $$\frac{dy}{dx}$$ out: $$ \frac{dy}{dx}(x - \frac{1}{x}) = -y $$ Then, divide by $$x-\frac{1}{x}$$ to isolate the $$\frac{dy}{dx}$$ term: $$ \frac{dy}{dx} = \frac{-y}{x - \frac{1}{x}} $$ This is the first order differential equation that represents the given family of curves.

Key concepts

Chain Rule

Understanding the chain rule is essential for differentiating composite functions, where one function is nested inside another. In the context of differential equations, this technique allows us to differentiate expressions where the variables are multiplicatively linked, such as in the equation e^xy.

When applying the chain rule to such an expression, we identify the 'outer' function (in this case, e^z, with z = xy) and the 'inner' function (z = xy). The derivative of the outer function with respect to the inner function is then multiplied by the derivative of the inner function with respect to the independent variable.

Here's the breakdown using the chain rule:

• Take the derivative of the outer function: d/dz(e^z) = e^z.
• Then, differentiate the inner function with respect to x: d/dx(xy) = y + x(dy/dx).
• Finally, multiply the two derivatives together to get the complete derivative with respect to x: e^xy(y + x(dy/dx)).

This approach is pivotal in calculating the derivative necessary for formulating a first order differential equation.

Product Rule for Differentiation

When we have a product of two functions that both depend on the same variable, we apply the product rule for differentiation to find the derivative. For example, if we need to differentiate uv, where both u and v are functions of x, the product rule tells us that the derivative of this product with respect to x is u'(the derivative of u with respect to x) times v, plus u times v'(the derivative of v with respect to x). In formal notation:(uv)' = u'v + uv'.

This rule turns out to be invaluable in working through the differentiation of the initial equation provided, where we treat c as a constant and y as a function of x. The product rule enables us to write the derivative of cy simply as c(dy/dx), as the derivative of a constant c with respect to x is zero.

Differential Equation Solution

The ultimate goal of this exercise is to arrive at a differential equation solution that accurately describes the relationship between the variables and their derivatives. To solve a differential equation means to find a function or a class of functions that satisfy the equation. In the case of the first-order differential equation derived from the family of curves e^xy = cy, the solution would yield y as a function of x that meets the original criteria.

These solutions often involve integrating factors, separable equations, or other methods such as undetermined coefficients. However, in the exercise at hand, isolating the derivative term (dy/dx) allows us to write the differential equation in a concise manner, providing a direct formula to correlate any change in x with the corresponding change in y, based on the given family of curves.

Isolating the Derivative Term

In the journey towards finding a solution for a differential equation, isolating the derivative term is a critical step. This involves manipulating the equation so that the term with dy/dx is by itself on one side of the equation. In terms of algebra, this involves combining like terms, factoring, and using basic operations to get the derivative dy/dx alone.

In our specific example, we began with a product of expressions and aimed to isolate dy/dx. We simplified the equation by dividing by terms that included x and then factored out the dy/dx from the remaining expression. After algebraic manipulation, we ended up with a clear, isolated derivative term on one side of the equation:

dy/dx = -y / (x - 1/x).

This clear formulation allows us to see directly how the gradient of the curve (the value of dy/dx) is influenced by the position on the curve (given by x and y), ultimately tying back to the original family of curves for which we sought a differential equation.