Question

Find all curves \(y=y(x)\) such that the tangent to the curve at any point \(\left(x_{0}, y\left(x_{0}\right)\right)\) intersects the \(y\) axis at \(y_{I}=x_{0}\).

Short answer

#tag_title# Step 2: Calculate the tangent line at a point (x0, y(x0)) #tag_content# In this step, we will calculate the tangent line at a point (x0, y(x0)). We will use the point-slope form of a tangent line, which is given by: \(y - y(x_0) = y'(x_0)(x - x_0)\) #tag_title# Step 3: Find the intersection point of the tangent line and the y-axis #tag_content# To find the intersection point of the tangent line and the y-axis, we must solve the tangent line equation for x = 0. Setting x = 0 in the tangent line equation, we obtain: \(y - y(x_0) = y'(x_0)(0 - x_0)\) Since the intersection point y-coordinate is equal to x0 (i.e., yI = x0), we can replace y with x0 in the equation and solve for y(x0): \(x_0 - y(x_0) = -y'(x_0) x_0\) #tag_title# Step 4: Solve the equation for y(x) #tag_content# In this step, we will solve the equation obtained in Step 3 for the curve y = y(x). Dividing both sides of the equation by x0 and rearranging the terms, we get: \(1 - \frac{y(x_0)}{x_0} = y'(x_0)\) Now, we can express this equation in terms of y(x) and y'(x): \(1 - \frac{y(x)}{x} = y'(x)\) This is a first-order differential equation for the curve y = y(x). Solving this equation will give us the family of curves that satisfy the given condition. Question: Find the family of curves y(x) such that their tangent lines at any point (x0, y(x0)) intersect the y-axis at yI = x0, given that the equation for the tangent line intersection is \(1 - \frac{y(x)}{x} = y'(x)\).

Step-by-step solution

Determine the derivative of the curve y(x) with respect to x

In this step, we will determine the derivative of the curve y=y(x) with respect to x. Let \(y'(x_0)\) be the derivative at \((x_0, y(x_0))\). Since we don't have a specific function for y(x) yet, we will denote its derivative as \(y'(x_0)\).

Key concepts

Tangent Intersection with Y-Axis

Understanding how a tangent to a curve interacts with the y-axis is a foundational concept in calculus. When a problem states that a tangent intersects the y-axis, it is referring to the point where the tangent line, if extended, would cross the y-axis. This is an essential point called the y-intercept of the tangent line.

In our exercise, we are given a unique condition for the family of curves we're trying to find: for any point \(x_0, y(x_0)\) on the curve, the tangent at that point intersects the y-axis at \(y_I = x_0\). This means the y-intercept of the tangent line equals the x-coordinate of the point of tangency. This condition will guide us to find a differential equation representing the curves.

Derivative of a Curve

The derivative of a curve at a given point represents the slope of the tangent line to the curve at that point. This concept is a crucial aspect of understanding the behavior of curves. In our problem, denoting the derivative by \(y'(x_0)\), we acknowledge it as the slope of the tangent line at the point \(x_0, y(x_0)\).

The derivative tells us how fast y is changing with respect to x. In graphical terms, a curve that is rapidly increasing will have a steep tangent line and hence a large derivative. Conversely, if the curve is flat, the tangent line is horizontal, and the derivative is zero. This insight is key to forming a differential equation that describes the family of curves we are seeking.

Implicit Differentiation

Implicit differentiation is utilized when dealing with equations that define y implicitly as a function of x, rather than explicitly. In our context, since we are not given an explicit function for \(y(x)\), we presume it is implicit and that we can still find its derivative with respect to x by applying implicit differentiation.

Implicit differentiation means taking the derivative of both sides of an equation with respect to x, treating y as a function of x. This way, even if y cannot be neatly expressed as \(y(x)\), we can still find \(y'\), the derivative of y with respect to x. This is particularly effective when the relationship between x and y is complex or when y is expressed in terms of x in a way that is difficult to solve for y explicitly.