Problem 4 - Differential Equations and Linear Algebra

Question

Solve the given differential equation. $$\frac{d y}{d x}=\frac{y}{x \ln x}$$

Step-by-Step Solution

Answer

The general solution of the given differential equation \(\frac{d y}{d x}=\frac{y}{x \ln x}\) is \(y = Ae^{\ln |\ln x|}\), with \(A = e^C\), an arbitrary constant.

Step 1: Rewrite the given differential equation as \(\frac{dy}{y}=\frac{dx}{x \ln x}\). Keep the \(y\)-terms on the left side and the \(x\)-terms on the right side. #Step 2: Integrate both sides of the equation#

Now, integrate both sides with respect to their respective variables: \(\int \frac{1}{y} dy = \int \frac{1}{x \ln x} dx\). #Step 3: Evaluate the integrals#

Step 2: The integral on the left side can be easily computed: \(\int \frac{1}{y} dy = \ln |y| + C_1\). Then, we need to find the value of the integral on the right side: \(\int \frac{1}{x \ln x} dx\). To solve this integral, use the substitution method: let \(u = \ln x\), so \(du = \frac{dx}{x}\). Our integral becomes: \(\int \frac{1}{u} du\), which, when evaluated, yields \(\ln |u| + C_2 = \ln |\ln x| + C_2\). Now, let's rewrite the equation with the calculated integrals: \(\ln |y| + C_1 = \ln |\ln x| + C_2\). #Step 4: Combine the constants and solve for y#

Combine the constants \(C_1\) and \(C_2\) into a single constant \(C\): \(\ln |y| = \ln |\ln x| + C\). To solve for \(y\), rewrite using exponentials: \(y = Ae^{\ln |\ln x|}\), where \(A = e^C\). The general solution of the given differential equation is: \(y = Ae^{\ln |\ln x|}\), with \(A = e^C\), an arbitrary constant.

Key Concepts

Separation of Variables

Separation of Variables is a fundamental method for solving differential equations. This technique involves rearranging a differential equation such that all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the opposite side. The key goal is to create a format where each side of the equation can be independently integrated.

For the differential equation \( \frac{dy}{dx} = \frac{y}{x \ln x} \), the separation of variables leads us to \( \frac{dy}{y} = \frac{dx}{x \ln x} \). Here, all \( y \) terms are isolated on the left, and all \( x \) terms are on the right. With this setup, each side can be integrated with respect to its respective variable, enabling us to easily progress towards a solution.

Ultimately, the concept of separation of variables simplifies the integration process, reducing complex relationships between variables to manageable integrals.

Integration Techniques

Integration Techniques are essential tools for solving separated equations. Once the equation is arranged into separate parts, integration is performed for both sides to find the general solution. In this particular exercise, integrating involves handling uncomplicated forms and applying basic integration knowledge.

For the left-hand side \( \int \frac{1}{y} \, dy \), integration is straightforward, resulting in \( \ln |y| \). On the right-hand side \( \int \frac{1}{x \ln x} \, dx \), the integral is more complex and requires substitution.

When both sides are integrated, they yield expressions that include constant terms \( C_1 \) and \( C_2 \). These constants arise because indefinite integrals can include any constant; they are crucial for forming the general solution of the differential equation.

Substitution Method

The Substitution Method is a clever integration technique used to simplify integrals that seem challenging at first glance. By substituting part of the integrand with a new variable, the expression becomes easier to integrate.

In this exercise, to solve \( \int \frac{1}{x \ln x} \, dx \), we use substitution: let \( u = \ln x \), implying \( du = \frac{1}{x} \, dx \). Substituting these into the integral simplifies it to \( \int \frac{1}{u} \, du \). This integral is much simpler and results in \( \ln |u| \).

Once evaluated, the substitution \( u = \ln x \) is reversed, leading to the expression \( \ln |\ln x| \). The Substitution Method, therefore, transforms a complex integral into a more approachable one, facilitating the solution process.