Problem 24 - A First Course in Differential Equations with Modeling Applications

Question

Verify that \(y_{1}(x)=x\) is a solution of \(x y^{\prime \prime}-x y^{\prime}+y=0\) Use reduction of order to find a second solution \(y_{2}(x)\) in the form of an infinite series. Conjecture an interval of definition for \(y_{2}(x)\)

Step-by-Step Solution

Answer

The second solution is \(y_2(x) = x \ln|x|\), defined on \((-\infty, 0) \cup (0, \infty)\).

Step 1: Verify given solution

First, we will differentiate the given function \(y_1(x) = x\). The first derivative is \(y_1'(x) = 1\) and the second derivative is \(y_1''(x) = 0\). Substitute these into the given differential equation \(x y'' - x y' + y = 0\):\[x \cdot 0 - x \cdot 1 + x = 0\]Simplifying gives:\[-x + x = 0\]Therefore, \(y_1(x) = x\) is indeed a solution of the differential equation.

Step 2: Setup for reduction of order

To find a second solution using reduction of order, assume a solution of the form \(y_2(x) = v(x) y_1(x) = v(x) x\), where \(v(x)\) is an unknown function to be determined. We need to find derivatives of \(y_2(x)\):The first derivative is:\[y_2'(x) = v'(x) x + v(x)\]The second derivative is:\[y_2''(x) = v''(x) x + 2v'(x)\]

Step 3: Substitute into the differential equation

Substitute \(y_2(x)\), \(y_2'(x)\), and \(y_2''(x)\) into the differential equation \(x y'' - x y' + y = 0\):\[x(v''(x) x + 2v'(x)) - x(v'(x) x + v(x)) + x v(x) = 0\]Simplify the expression:\[x^2 v''(x) + 2x v'(x) - x^2 v'(x) - x v(x) + x v(x) = 0\]Cancel terms and simplify further:\[x^2 v''(x) + x v'(x) = 0\]

Step 4: Solve the simplified equation

The equation \(x^2 v''(x) + x v'(x) = 0\) can be solved by separation of variables or by recognizing it as a first-order linear homogeneous differential equation. Divide through by \(x^2\):\[v''(x) + \frac{v'(x)}{x} = 0\]This suggests \(v'(x) = \frac{C}{x}\) for some constant \(C\). Integrating, we have:\[v(x) = C \ln|x| + D\]

Step 5: Formulate second solution

Using \(y_2(x) = v(x) x\), we substitute \(v(x)\) back in:\[y_2(x) = (C \ln|x| + D)x\]Since we are looking for a fundamentally different solution, and we already know one solution is \(x\), we take \(C=1\) and \(D=0\), resulting in:\[y_2(x) = x \ln|x|\]

Step 6: Determine interval of definition

The form of \(y_2(x) = x \ln|x|\) suggests the solution is defined wherever \(\ln|x|\) is defined, which is for \(x > 0\) or \(x < 0\). Therefore, the interval of definition for \(y_2(x)\) is \((-\infty, 0) \cup (0, \infty)\), excluding the point where \(x = 0\) because the natural logarithm is not defined there.

Key Concepts

Differential Equations

Differential equations play a crucial role in various fields of science and engineering. These are mathematical equations that involve functions and their derivatives. The core idea is to describe how a function changes, often representing real-world phenomena like growth rates, motion, or heat transfer.

A typical differential equation has the form: \ \( F(x, y, y', y'', \ldots) = 0 \), where \(y = f(x)\) is a function of \(x\) and \(y', y''\) are its first and second derivatives, respectively.

There are multiple types of differential equations:

• **Ordinary Differential Equations (ODEs)** - involve a single independent variable and its derivatives.
  
• **Partial Differential Equations (PDEs)** - involve multiple independent variables and partial derivatives.

In solving differential equations, the goal is to find a function or a set of functions that satisfy the equation; simply put, these solutions describe the behavior dictated by the equation.

Knowing how to solve these equations is essential because they model how systems evolve over time. The example provided illustrates a linear second-order differential equation. It's linear because each term is either a constant or a product of a constant and a derivative, and second-order because the highest derivative is the second derivative.

Infinite Series Solution

An infinite series solution is a method used to express the solution to a differential equation as a sum of an infinite series of terms. This approach is particularly useful when solving equations where simple algebraic solutions are impossible or impractical.

In the exercise, the method of **reduction of order** is used to find a second solution to the differential equation starting with a known solution. This method often involves:

• Assuming a solution in the form \(y_2(x) = v(x) y_1(x)\), where \(v(x)\) is an unknown function.
  
• Using the initial solution, here \(y_1(x) = x\), to simplify the differential equation.
  
• Deriving a new simpler equation for \(v(x)\) and solving it, often resulting in a series or complex function.

In our case, after substituting and simplifying, we derive the solution in terms of logarithmic functions, leading to the expression \(y_2(x) = x \ln|x|\).

This infinite series technique is powerful as it accommodates complex differentials, opening paths to approximate solutions for practical applications like physics simulations or financial models.

Interval of Definition

The interval of definition in a differential equation refers to the range of the independent variable over which the solution is valid.

For solutions involving logarithmic or fractional terms, the interval of definition helps identify where these terms are valid.

In the given exercise, after deriving \(y_2(x) = x \ln|x|\), we consider where \(\ln|x|\) is defined. The function \(\ln(x)\) is defined only for positive values of \(x\). However, since the expression includes \(|x|\), it expands to both negative and positive \(x\) values, excluding zero.

The logical breakdown is:

• **\(x > 0\):** \(\ln|x|\) is defined for all positive values.
  
• **\(x < 0\):** Absolute value ensures \(\ln|x|\) operates over all negative values, effectively converted to positive.
  
• **\(x = 0\):** Not included, as \(\ln(0)\) is undefined.

Thus, the complete interval of definition is \((-\infty, 0) \cup (0, \infty)\), showcasing the solution's applicability over the entire real number line, except for \(x = 0\). This consideration is vital for ensuring the solution's applicability and reliability in practical scenarios.