Problem 33 - Differential Equations and Linear Algebra

Question

Let \(F_{1}\) and \(F_{2}\) be two families of curves with the property that whenever a curve from the family \(F_{1}\) intersects one from the family \(F_{2},\) it does so at an angle \(a \neq \pi / 2 .\) If we know the equation of \(F_{2},\) then it can be shown (see Problem 23 in Section 1.1 ) that the differential equation for determining \(F_{1}\) is $$ \frac{d y}{d x}=\frac{m_{2}-\tan a}{1+m_{2} \tan a} $$ where \(m_{2}\) denotes the slope of the family \(F_{2}\) at the point \((x, y)\) Use Equation \((1.8 .16)\) to determine the equation of the family of curves that cuts the given family at an angle \(\alpha=\pi / 4\) $$x^{2}+y^{2}=c$$

Step-by-Step Solution

Answer

The equation for the family of curves \(F_1\) that intersect the given family \(F_2: x^2+y^2=c\) at an angle \(\alpha=\pi/4\) is: \(y = 3x + 2x \ln|x+y| - k\)

Step 1: Find the slope of the family of curves F_2 at the point (x, y)

The given equation for the family of curves \(F_2\) is: \(x^2 + y^2 = c\) To find the slope at a point \((x, y)\), we need to differentiate the equation with respect to x: \(\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(c)\) The left side gives us: \(2x + 2y \frac{dy}{dx} = 0\) Now, we solve for \(\frac{dy}{dx}\): \(\frac{dy}{dx} = -\frac{x}{y}\) The slope of the family of curves \(F_2\) at the point \((x, y)\) is given by \(m_2 = -\frac{x}{y}\).

Step 2: Plug the slope from Step 1 into the differential equation

Now we will substitute the slope we found in Step 1 (i.e., \(m_2 = -\frac{x}{y}\)) into the differential equation from the problem statement: \(\frac{dy}{dx} = \frac{m_2 - 1}{1 + m_2}\) Substituting the value of \(m_2\), we get: \(\frac{dy}{dx} = \frac{-\frac{x}{y} - 1}{1 - \frac{x}{y}}\)

Step 3: Simplify and solve the differential equation

Rearrange the expression in the denominator to obtain: \(\frac{dy}{dx} = \frac{-x - y}{y - x}\) Now we need to solve this first-order differential equation. It is a separable equation, so we will separate the variables and then integrate: \(\frac{y - x}{x + y} dy = dx\) Now, integrate both sides: \(\int \frac{y - x}{x + y} dy = \int dx\) Let u = x+y: \(\int \frac{u - 2x}{u} du = \int dx\) \(\int (\frac{u}{u} - \frac{2x}{u}) du = \int dx\) \(\int (1 - \frac{2x}{x+y}) du = \int dx\) \(\int 1 du - \int \frac{2x}{x+y} du = \int dx\) \(u - 2x \int \frac{1}{u} du = x + k\) Now, substituting back u = x+y: \(x + y - 2x \int \frac{1}{x+y} dy = x + k\) \(-x + y - 2x \ln|x + y| = x + k\) \(y = 3x + 2x \ln|x+y| - k\) Finally, we have obtained the equation for the family of curves \(F_1\) which intersect the given family \(F_2\) at an angle \(\alpha = \pi/4\): \(y = 3x + 2x \ln|x+y| - k\)

Key Concepts

Family of Curves

A family of curves is a set of curves that can be represented by a single equation with one or more parameters. These parameters can be varied to generate different curves within the family. In the context of differential equations, a family of curves can be used to identify solutions that share a common structural form. When solving problems involving two intersecting families of curves, such as the ones in this exercise, understanding how these families are linked through differential equations is crucial.

• Each curve in a family is a potential solution parameterized by a constant.
• The family of curves of interest here is described by the relationship between two dependent variables, typically x and y.
• Parameters can be changed to obtain different curves within the family, highlighting the diversity yet coherence within the set.

In our exercise, we dealt with families of circles given by the equation \(x^2 + y^2 = c\). Differentiating this helps us explore how one line from Family F1 intersects another from Family F2, sharing common traits of their families such as parallel tangents.

Slope of a Curve

The slope of a curve at a given point is a measure of the incline or steepness at that point and is described by the derivative of the curve's equation. For curves given in Cartesian coordinates, the differential \(\frac{dy}{dx}\) represents this slope. Finding the slope is a fundamental step in understanding how curves behave, especially when analyzing intersections or tangent lines.

• Calculation of the slope involves differentiating the equation of the curve.
• The differential provides a linear approximation of the curve's behavior locally at a point.
• The slope is crucial in determining the behavior of intersecting curves, informing angles and intersections.

In this exercise, for the family \(F_2\) given by \(x^2 + y^2 = c\), we find the slope as \(m_2 = -\frac{x}{y}\). Knowing \(m_2\) is essential as it directly affects the angle of intersection and subsequent calculations.

Separable Differential Equations

Separable differential equations are a class of first-order differential equations that can be transformed by algebraic manipulation into a set where all terms containing one variable are on one side of the equation and all terms containing another variable are on the opposite side. This separation allows us to solve them more easily through integration.

• The process involves rearranging the equation to isolate terms of different variables on separate sides.
• Once separated, integration is performed on both sides to solve the equation.
• Separable equations are particularly convenient because they convert a complex problem into a series of simpler calculations.

In our solution, we addressed the differential equation \(\frac{dy}{dx} = \frac{-x-y}{y-x}\) by separating variables, executing integration, and reshaping it into our target equation.

Integration

Integration is the process of finding the function given its derivative, a key tool in solving differential equations. Once a differential equation is separated, integration helps find the solution or family of solutions that fit the given conditions.

• Indefinite integrals provide general solutions, usually with a constant \(k\) representing unknown initial conditions.
• Integration can often involve substitutions to simplify the function we're integrating.
• Employing integration by parts or substitution can be effective in complex scenarios.

During the solution, we separated the equation to focus on \(\int \frac{y-x}{x+y}dy = \int dx\), allowing integration to yield the final relationship between \(x\) and \(y\). This integral approach was crucial in arriving at the expression \(y = 3x + 2x \ln|x+y| - k\).

Angle of Intersection

The angle of intersection is the angle formed between two curves where they intersect. It is directly related to the slopes of the tangent lines to the curves at their point of intersection. Knowing this angle is essential when determining the interaction between families of curves.

• Two curves intersect at an angle other than \(90^\circ\) when their tangent slopes do not result in perpendicular scalars.
• The formula \(\tan \alpha = \frac{|m_1 - m_2|}{1 + m_1 m_2}\) computes the angle of intersection \(\alpha\), where \(m_1\) and \(m_2\) are the slopes of the curves.
• The angle provides insight into how curves behave in relation to each other, guiding the solution of differential equations.

For the curves in this exercise, setting \(\alpha = \frac{\pi}{4}\) helped derive the differential equation, leading to the solution that describes how the family \(F_1\) intersects \(F_2\) at the specified angle.