Problem 30 - Differential Equations and Linear Algebra

Question

Use the technique derived in the previous problem to solve the given differential equation. $$y^{\prime}+x^{-1} y=\cos x, \quad x > 0$$

Step-by-Step Solution

Answer

The final form of the solution to the given differential equation is \(y(x) = \frac{1}{x}(\int x \cos x dx + C)\), where \(C\) is the constant of integration and the integral represents an indefinite integral.

Step 1: Identify the differential equation as first-order linear

The given differential equation is of the form: \[y' + \frac{1}{x} y = \cos x, \quad x > 0\] This is a first-order linear ordinary differential equation with the function \(p(x) = \frac{1}{x}\) and the inhomogeneous term \(f(x) = \cos x\).

Step 2: Find the integrating factor

The integrating factor, denoted by \(I(x)\), is given by the formula \(I(x) = e^{\int p(x) dx}\). For our problem, we need to find the integrating factor: \[I(x) = e^{\int \frac{1}{x} dx}\] Integrating, we get: \[I(x) = e^{\ln{x}}\] Therefore, the integrating factor is: \[I(x) = x\]

Step 3: Multiply both sides of the equation by the integrating factor

Now we will multiply both sides of our differential equation by the integrating factor \(I(x) = x\). This will give us: \[x \cdot (y' + \frac{1}{x} y) = x \cos x\] Simplifying, we get: \[x y' + y = x \cos x\]

Step 4: Rewrite the equation in terms of a single derivative

The left side of the equation can be rewritten as a single derivative, which is the derivative of the product of \(y(x)\) and the integrating factor \(I(x) = x\): \[\frac{d}{dx}(xy) = x \cos x\]

Step 5: Integrate both sides of the equation

Now we will integrate both sides of the equation with respect to \(x\): \[\int \frac{d}{dx}(xy) dx = \int x \cos x dx\] The left side of the equation can be easily integrated to obtain: \[xy = \int x \cos x dx + C\] where \(C\) is the constant of integration.

Step 6: Solve for \(y(x)\)

Finally, we need to solve for the function \(y(x)\): \[y(x) = \frac{1}{x}(\int x \cos x dx + C)\] Unfortunately, there is no elementary function that can be expressed as the antiderivative of \(x \cos x\). Therefore, the final form of the solution is: \[y(x) = \frac{1}{x}(\int x \cos x dx + C)\] where \(C\) is the constant of integration and the integral represents an indefinite integral.

Key Concepts

Integrating Factor

In first-order linear differential equations, the integrating factor is a powerful tool used to simplify the problem. It transforms a differential equation into a form that can be easily integrated. Briefly put, the integrating factor is a function that you multiply with a given differential equation to make it easier to solve.
The formula for the integrating factor, denoted as \( I(x) \), is given by \( I(x) = e^{\int p(x) \: dx} \), where \( p(x) \) is the coefficient of \( y \) in the standard form of the equation \( y' + p(x) y = f(x) \).

In our example:

• The function \( p(x) \) is \( \frac{1}{x} \).
• Calculating the integrating factor involves integrating \( \frac{1}{x} \), which results in \( \ln{x} \).
• Substituting back, the integrating factor is \( I(x) = e^{\ln{x}} = x \).

This integrating factor simplifies the equation, setting the stage for straightforward integration in subsequent steps.

Inhomogeneous Differential Equation

An inhomogeneous differential equation contains a non-zero term on one side, different from homogeneous equations, which equalize to zero. These inhomogeneous or non-homogeneous equations typically take the form \( y' + p(x) y = f(x) \), where \( f(x) \) is not zero.

Here's what you need to understand about these equations:

• They contain a function \( f(x) \), e.g., in our case, \( f(x) = \cos{x} \).
• This non-zero term often represents an external source or input.
• Solving them involves finding a particular solution that satisfies \( f(x) \) and the homogeneous solution to \( y' + p(x) y = 0 \).

By using techniques like integrating factors, one can systematically manage these equations to find solutions that encompass both particular and homogeneous solutions.

Technique of Integration

The technique of integration is crucial in solving differential equations once the equation has been transformed using an integrating factor. After applying this factor, the left side of the equation often ends up being the derivative of a product of two functions.
For the problem we're looking at, the left-hand side of the equation simplifies to \( \frac{d}{dx}(xy) \) after the integrating factor \( x \) is applied.

Here's a step-by-step idea of how integration helps:

• Recast the equation such that it's ready to be integrated.
• Integrate both sides with respect to \( x \).
• Don't forget the constant of integration, \( C \), which represents indefinite integration and varies based on initial conditions.

Note that some functions, like \( \int x \cos x \, dx \) in our exercise, don't have neat elementary antiderivatives. However, setup-wise, the process remains consistent, allowing for a structured path to finding \( y(x) \). This method is foundational to solving many first-order linear differential equations.