Question

The method outlined in Problem 30 can be used for any homogeneous equation. That is, the substitution \(y=x v(x)\) transforms a homogeneous equation into a separable equation. The latter equation can be solved by direct integration, and then replacing \(v\) by \(y / x\) gives the solution to the original equation. In each of Problems 31 through 38 : $$ \begin{array}{l}{\text { (a) Show that the given equation is homogeneous. }} \\\ {\text { (b) Solve the differential equation. }} \\ {\text { (c) Draw a direction field and some integral curves. Are they symmetric with respect to }} \\ {\text { the origin? }}\end{array} $$ $$ \frac{d y}{d x}=\frac{x+3 y}{x-y} $$

Short answer

The short answer to the given question is: (a) The given differential equation is homogeneous since the degrees of the terms in the numerator and denominator are the same (2). (b) Using the substitution \(y=xv(x)\), we solve the transformed equation and get the solution: $$ y = \frac{3x-x^3e^{C'}}{2} $$ (c) The direction field and integral curves can be drawn using software like Desmos or Mathematica. These plots help visualize the behavior of the function and its symmetry. In this case, the integral curves are symmetric with respect to the origin.

Step-by-step solution

Show that the given equation is homogeneous

We are given the differential equation: $$ \frac{dy}{dx} = \frac{x+3y}{x-y} $$ To show that it is homogeneous, we need to check if the degree of terms in the numerator and the denominator are the same. Here, assuming \(x\) and \(y\) have a degree of 1: $$ \text{Degree of the numerator} = \text{Degree}(x) + \text{Degree}(3y) = 1 + 1 = 2 \\ \text{Degree of the denominator} = \text{Degree}(x) + \text{Degree}(y) = 1 + 1 = 2 $$ Since the degrees of the numerator and the denominator are the same (2), the equation is homogeneous.

Use the substitution \(y=xv(x)\)

Now we will use the substitution \(y=xv(x)\) to transform the homogeneous equation into a separable equation. Differentiate both sides with respect to \(x\) to find \(\frac{dy}{dx}\): $$ \frac{dy}{dx} = v(x) + x\frac{dv(x)}{dx} $$ Now replace \(y\) and \(\frac{dy}{dx}\) in the original equation: $$ v(x) + x\frac{dv(x)}{dx} = \frac{x+3xv(x)}{x-xv(x)} $$

Solve the transformed separable equation

Now we have a separable equation, separate the variables and integrate both sides: $$ \frac{v(x)+x\frac{dv(x)}{dx}}{1+3v(x)-v^2(x)} = \frac{dx}{x} $$ Integrate both sides: $$ \int \frac{v(x)+x\frac{dv(x)}{dx}}{1+3v(x)-v^2(x)} dv(x) = \int \frac{1}{x} dx $$ The integral on the left side requires substitution. Let \(u = 1+3v(x)-v^2(x)\). Then, \(\frac{du}{dx} = 3\frac{dv}{dx}-2v(x)\frac{dv}{dx}\). Simplifying the integral and integrating both sides: $$ \int \frac{1}{3-2v(x)}du = \int \frac{1}{x} dx \\ \frac{1}{2}\ln|3-2v(x)| = \ln|x| + C $$ $$ \ln|3-2v(x)| = 2\ln|x| + C' $$ Take the exponent of both sides and solve for \(v(x)\): $$ 3-2v(x) = x^2e^{C'} \\ v(x) = \frac{3-x^2e^{C'}}{2} $$

Replace \(v\) with \(y/x\) and find the solution to the original equation

Now replace \(v\) with \(y/x\) to find the solution to the original equation: $$ \frac{y}{x} = \frac{3-x^2e^{C'}}{2} $$ Multiply both sides by \(x\) to get \(y\): $$ y = \frac{3x-x^3e^{C'}}{2} $$

Discuss the direction field, integral curves and symmetry

The direction field for the given equation can be drawn using software like Desmos or Mathematica, and the integral curves can be found by plotting solutions. The plot of the direction field will help us visualize the behavior of the function as well as its symmetry. In this case, we can observe that the integral curves are symmetric with respect to the origin, since the substitution \(y = xv(x)\) preserves the degrees of the terms, which represent the 'shape' of the function.

Key concepts

Degree of Differential Equations

In the realm of differential equations, understanding the concept of degree is essential. The degree of a differential equation is defined as the highest power of the highest derivative when the equation is polynomial in derivatives. For a given differential equation, \( \frac{dy}{dx} = \frac{x+3y}{x-y} \), checking its degree involves examining both the numerator and the denominator. Here, each variable, \(x\) and \(y\), is considered to have a degree of 1.

• Numerator: The degree comes from \(x\) and \(3y\), both having a degree of 1, resulting in a total degree of 2 (1 + 1).
• Denominator: Similarly, \(x\) and \(y\) each contribute a degree of 1, also summing to 2.

Since both parts have the same degree, it confirms that the differential equation is homogeneous. This property is pivotal because it implies that the equation can be solved using specific techniques such as substitution, which simplifies the process.

Separable Equations

A separable differential equation is one that can be rewritten to allow the variables to be separated on opposite sides of the equation. Transforming a homogeneous equation into a separable form lets us handle different variables independently, making integration straightforward. In the exercise, we show that by using the substitution \(y = xv(x)\), we transform the original equation into a form that is separable. Here’s how it works:
1. Substitute \(y = xv(x)\) into the original equation to express \(y\) in terms of \(x\) and \(v(x)\).2. Differentiate the substitution \(y = xv(x)\) with respect to \(x\) to find \(\frac{dy}{dx}\).3. This yields: \(\frac{dy}{dx} = v(x) + x\frac{dv(x)}{dx}\).4. Plugging these into the original differential equation, we reorganize terms to achieve a separable form.Once separated, both sides can be integrated independently, which is much simpler to solve. The key bonus is that separable equations are significantly easier to integrate, thanks to their straightforward separation of variables.

Method of Substitution in Differential Equations

The method of substitution is a technique used to solve complex differential equations by introducing a new variable that simplifies the equation. Specifically, for homogeneous equations, using the substitution \(y = xv(x)\) turns them into a separable form. Here’s the explanation in detail:

• By introducing \(v(x)\), the dependency between \(y\) and \(x\) can be expressed in a single term.
• This substitution helps simplify the compound structure of a homogeneous equation into a more manageable form.
• For the given exercise, the substitution transforms the equation \(\frac{dy}{dx} = \frac{x+3y}{x-y}\) into conditions that are separable by isolating \(v(x)\).

Once the homogeneous equation is transformed, it becomes easier to handle, allowing straightforward integration to find \(v(x)\). The last step involves reversing the substitution to achieve the solution in terms of the original variables. This method simplifies solving homogeneous differential equations and highlights the power of substitution as a problem-solving gateway.