Question

Find a first order differential equation for the given family of curves. $$ y=e^{x^{2}}+c e^{-x^{2}} $$

Short answer

Question: Find the first order differential equation representing the family of curves given by the equation \(y=e^{x^{2}}+ce^{-x^{2}}\), where \(c\) is an arbitrary constant. Answer: \(\frac{dy}{dx} = 4x e^{x^{2}} - 2xy\)

Step-by-step solution

Differentiating the equation with respect to x

First, let's differentiate the given equation with respect to \(x\). We will use the Chain Rule to differentiate both terms of the equation. $$ \frac{dy}{dx} = \frac{d}{dx} (e^{x^{2}}) + \frac{d}{dx} (c e^{-x^{2}}) $$ Differentiating both terms, we get: $$ \frac{dy}{dx} = 2x e^{x^{2}} - 2cx e^{-x^{2}} $$

Eliminate the constant c

In order to eliminate the constant \(c\), we want a relationship involving \(y\), \(x\), and \(dy/dx\). From the given equation of family of curves, we can write \(ce^{-x^{2}}\) as: $$ ce^{-x^{2}} = y - e^{x^{2}} $$ Now, multiply both sides of the equation by \(-2x\), we get: $$ -2cx e^{-x^{2}} = -2x(y - e^{x^{2}}) $$ Note that the left side of this equation has the same form as the second term of the differentiated equation we found in step 1.

Substitute the relationship of c back into the differentiated equation

Now, we can substitute the expression of \(-2cx e^{-x^{2}}\) we found in step 2 into the differentiated equation from step 1: $$ \frac{dy}{dx} = 2x e^{x^{2}} - 2x(y - e^{x^{2}}) $$

Simplifying the equation

Finally, simplify the first order differential equation: $$ \frac{dy}{dx} = 2x e^{x^{2}} - 2x(y - e^{x^{2}}) $$ Distribute the term \(-2x\): $$ \frac{dy}{dx} = 2x e^{x^{2}} - 2xy + 2x e^{x^{2}} $$ Combine the terms involving \(e^{x^{2}}\): $$ \frac{dy}{dx} = 4x e^{x^{2}} - 2xy $$ This is the first order differential equation representing the given family of curves.

Key concepts

Differentiation

Differentiation is the process of calculating the derivative of a function. It measures how a function's output value changes as its input value changes. In simpler terms, the derivative tells us the rate at which something is changing. For example, if we consider the function representing the position of a moving object over time, its derivative would provide us with the object's velocity, which is the rate of change of position with respect to time.

In the context of our exercise, differentiation is applied to the function \( y = e^{x^{2}} + ce^{-x^{2}} \) to find its derivative with respect to \( x \). This process involves the rate at which \( y \) changes as \( x \) changes, which is a crucial step in finding the first order differential equation for the given family of curves.

Chain Rule

The chain rule is a fundamental differentiation rule in calculus used to compute the derivative of the composition of two or more functions. It states that if a variable \( y \) depends on a variable \( u \) which, in turn, depends on a variable \( x \), then the derivative of \( y \) with respect to \( x \) can be found by multiplying the derivative of \( y \) with respect to \( u \) and the derivative of \( u \) with respect to \( x \). In mathematical terms, \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

In our exercise, the chain rule is used to differentiate the terms \( e^{x^{2}} \) and \( ce^{-x^{2}} \), where an intermediary variable like \( u = x^2 \) is implicitly involved. This allows us to handle more complex functions like exponentials of quadratics efficiently, which is often encountered in first order differential equations.

Family of Curves

A family of curves represents a set of curves that are described by a given equation with one or more arbitrary constants. In our exercise, \( y = e^{x^{2}} + ce^{-x^{2}} \) represents a family of curves where \( c \) is a constant that can take any value. Each value of \( c \) gives us a different curve in this family.

When dealing with a family of curves in the context of differential equations, one usually aims to find a differential equation that holds true for all members of the family. This often involves eliminating the arbitrary constants from the differential expression, to find a single equation that characteristically defines the whole family without dependence on specific constants.

Explicit Solutions

Explicit solutions of differential equations are solutions where the dependent variable is given directly in terms of the independent variable, without any implicit relationships or parameters left to be determined. They are in the form \( y = f(x) \), which makes them straightforward to analyze and graph.

However, in our textbook exercise, we are not finding an explicit solution. Instead, we are obtaining a differential equation from a family of curves. Yet, recognizing explicit solutions can be beneficial for students as it helps in understanding what the end result of solving a differential equation typically looks like, even though in some complex situations, such simple forms may not be attainable, and numerical or graphical methods might be necessary to understand the behavior of the solutions.