Problem 8 - Differential Equations and Linear Algebra

Question

Solve the given differential equation. $$y^{\prime}+\frac{1}{x \ln x} y=9 x^{2}$$

Step-by-Step Solution

Answer

The general solution for the given differential equation is: \(y(x) = \frac{1}{e^{Ei(\ln x)}} \int 9x^2 e^{Ei(\ln x)} dx + \frac{C}{e^{Ei(\ln x)}}\), where \(Ei\) is the Exponential Integral function.

Step 1: Identify the given differential equation

The given differential equation is of the form: \(y'(x) + P(x)y(x) = Q(x)\) Here, \(P(x) = \frac{1}{x \ln x}\) and \(Q(x) = 9x^2\).

Step 2: Find the integrating factor

To find the integrating factor (IF), we need to integrate \(P(x)\) with respect to x: \(IF(x) = e^{\int P(x) dx} = e^{\int \frac{1}{x \ln x} dx}\) To compute this integral, let's use the substitution method. Let \(u = \ln x\), so \(x = e^u\), and \(dx = e^u du\): \(IF(x) = e^{\int \frac{1}{u} e^u du}\) Integrating with respect to u: \(IF(x) = e^{\int \frac{e^u}{u} du}\) Here, we can't find a closed-form expression for this integral. It's actually the definition of the Exponential Integral (Ei). So, let's represent it with Ei: \(IF(x) = e^{Ei(u)} = e^{Ei(\ln x)}\)

Step 3: Multiply the equation with the integrating factor

Now, let's multiply the given differential equation with the integrating factor: \(e^{Ei(\ln x)}(y'(x) + \frac{1}{x \ln x} y(x)) = 9x^2 e^{Ei(\ln x)}\)

Step 4: Integrate both sides of the equation

Now, the left side of the equation is an exact derivative, so integrating both sides with respect to x: \(\int e^{Ei(\ln x)}(y'(x) + \frac{1}{x \ln x} y(x)) dx = \int 9x^2 e^{Ei(\ln x)} dx\) The derivative of y(x) times the integrating factor is: \(\frac{d}{dx}(y(x) e^{Ei(\ln x)})\) Integrating both sides with respect to x, we get: \(y(x) e^{Ei(\ln x)} = \int 9x^2 e^{Ei(\ln x)} dx + C\)

Step 5: Solve for y(x)

To find y(x), divide the equation by \(e^{Ei(\ln x)}\): \(y(x) = \frac{1}{e^{Ei(\ln x)}} \int 9x^2 e^{Ei(\ln x)} dx + \frac{C}{e^{Ei(\ln x)}}\) Unfortunately, we can't find an exact closed-form expression for the integral in this case. However, we can express the solution in terms of the integral and the Exponential Integral function. So, the general solution for the given differential equation is: \(y(x) = \frac{1}{e^{Ei(\ln x)}} \int 9x^2 e^{Ei(\ln x)} dx + \frac{C}{e^{Ei(\ln x)}}\) This is the answer in terms of Exponential Integral function and integral representation.

Key Concepts

Integrating Factor

An integrating factor is a crucial tool for solving linear first-order differential equations. The idea is to multiply both sides of the differential equation by this factor, which simplifies the equation into an exact derivative. It helps us by transforming a non-exact differential equation into an exact one, making it easier to integrate.

In the context of the exercise, the integrating factor is found by using the expression:

• \( IF(x) = e^{\int P(x) dx} \)

Here, \( P(x) \) was identified as \( \frac{1}{x \ln x} \), which after integration with respect to x, contributed to defining the integrating factor.

The integrating factor helps reformulate the given differential equation into a form where the left side becomes the derivative of a product function. This methodology allows for straightforward integration and simplifies the complex original equation.

Exponential Integral

The Exponential Integral, often denoted as \( Ei(x) \), is a special integral that does not have a simple closed-form expression. It comes into play whenever an integration cannot be expressed in terms of elementary functions. In this exercise's context, the appearance of the expression \( e^{Ei(\ln x)} \) arises during the integration process of the integrating factor.

While it may seem mysterious at first, the Exponential Integral is a well-studied mathematical function and is especially useful in applied mathematics, physics, and engineering. When solutions can only be expressed in terms of \( Ei(x) \), it usually indicates a higher complexity or an advanced integration path.

Within the exercise, because a direct integration of the form \( \int \frac{e^u}{u} du \) is not possible simply, we replace it using \( Ei(u) \). This transitions the problem into one where further calculations may involve numerical methods or software that can handle the Exponential Integral.

Substitution Method

The substitution method is a valuable technique used to simplify integration problems, making them more manageable. In this exercise, we used substitution to transform a complex integrand into something easier to integrate. The key idea is to choose a substitution variable that can simplify the differential equation or integral.

Here, the substitution \( u = \ln x \) was made to handle the integral \( \int \frac{1}{x \ln x} dx \). This choice makes sense because differentiating \( \ln x \) results in \( \frac{1}{x} \), effectively reducing the complexity of the integrand. With \( u = \ln x \), and thus \( x = e^u \) and \( dx = e^u du \), the integral becomes \( \int \frac{e^u}{u} du \), which is expressed in terms of the Exponential Integral.

Using the substitution method effectively bridges the gap between an impossible-to-integrate expression and a more familiar form, making the solving of differential equations with auxiliary techniques like the integrating factor more straightforward.