Problem 13 - Advanced Engineering Mathematics

Question

For \(y^{\prime}+\left(1+\frac{2}{x}\right) y=\frac{e^{x}}{x^{2}}\) an integrating factor is \(e^{\int[1+(2 / x)] d x}=x^{2} e^{x}\) so that \(\frac{d}{d x}\left[x^{2} e^{x} y\right]=e^{2 x}\) and \(y=\frac{1}{2} \frac{e^{x}}{x^{2}}+\frac{c e^{-x}}{x^{2}}\) for \(0< x<\infty .\) The transient term is \(\frac{c e^{-x}}{x^{2}}\).

Step-by-Step Solution

Answer

The transient term is \(\frac{Ce^{-x}}{x^{2}}\).

Step 1: Identify the Equation

We are given the differential equation: \[ y' + \left(1+\frac{2}{x}\right) y = \frac{e^{x}}{x^{2}} \]This is a first-order linear differential equation.

Step 2: Find the Integrating Factor

The given integrating factor for the equation is:\[ e^{\int[1+(2 / x)] \, dx} = x^{2}e^{x} \] We need this to multiply the entire differential equation to make it exact.

Step 3: Multiply by the Integrating Factor

Multiply the entire differential equation by the integrating factor \(x^{2}e^{x}\):\[ x^{2}e^{x} \left( y' + \left(1+\frac{2}{x}\right) y \right) = x^{2}e^{x} \cdot \frac{e^{x}}{x^{2}} \] Simplifying the equation gives us:\[ \frac{d}{dx}\left[x^{2}e^{x}y\right] = e^{2x} \]

Step 4: Integrate Both Sides

To solve for \(y\), integrate both sides with respect to \(x\):\[ \int \frac{d}{dx}\left[x^{2}e^{x}y\right] \, dx = \int e^{2x} \, dx \]This results in:\[ x^{2}e^{x}y = \frac{1}{2}e^{2x} + C \] where \(C\) is the constant of integration.

Step 5: Solve for y

Now, isolate \(y\) to solve the equation:\[ y = \frac{1}{x^{2}e^{x}} \cdot \left(\frac{1}{2}e^{2x} + C\right) \]Simplifying this gives:\[ y = \frac{1}{2}\frac{e^{x}}{x^{2}} + \frac{Ce^{-x}}{x^{2}} \]

Step 6: Identify the Transient Term

Compare the derived solution for \(y\) with the given form. The expression:\[ \frac{Ce^{-x}}{x^{2}} \]represents the transient term. It diminishes as \(x\) becomes very large.

Key Concepts

Integrating Factor

When dealing with first-order linear differential equations, one powerful tool to solve them is the integrating factor. This special function helps us transform a non-exact differential equation into an exact one, making it much easier to solve.
To find the integrating factor, we often look for a function that will multiply across the entire equation, turning its left-hand side into the derivative of some function. In our exercise, the given differential equation was:
\[ y' + \left(1+\frac{2}{x}\right)y = \frac{e^{x}}{x^{2}} \]
Here, the integrating factor is calculated as:
\[ e^{\int[1+(2 / x)] \, dx} = x^{2}e^{x} \]
This function, \(x^{2}e^{x}\), when multiplied with the entire equation, converts it into a form that becomes easily integrable.
It's important to note that the integrating factor is always derived from the coefficient of \(y\) in the equation, ensuring that the left-hand side is an exact derivative.

Transient Term

In differential equations, especially in contexts dealing with physical or engineering problems, a transient term refers to a part of the solution that eventually disappears or decays over time or as the variable increases.
In the provided solution,
\[ y = \frac{1}{2}\frac{e^{x}}{x^{2}} + \frac{Ce^{-x}}{x^{2}} \]
the transient term is identified as:
\[ \frac{Ce^{-x}}{x^{2}} \]
This part of the solution diminishes when \(x\) grows larger, particularly because \(e^{-x}\) decreases towards zero as \(x\) increases. As a result, the transient term becomes negligible, and what remains is the steady-state solution, \(\frac{1}{2}\frac{e^{x}}{x^{2}}\).
Transient terms are crucial in understanding how solutions to differential equations behave in the long run and can often indicate the stability of a system or process.

Exact Differential Equation

The term "exact differential equation" refers to an equation that can be expressed in a form where the left-hand side is the exact derivative of some function. This simplification allows us to integrate and solve the equation directly.
In our context, after incorporating the integrating factor, the differential equation was turned into an exact form:
\[ \frac{d}{dx}\left[x^{2}e^{x}y\right] = e^{2x} \]
Here, the left-hand side is already the derivative of \(x^{2}e^{x}y\) with respect to \(x\), making the problem simpler to manage.
By transforming a differential equation into an exact one, solving it becomes a matter of straightforward integration.
This conversion to an exact form is essential for eliminating complexity and effectively reaching the solution.