Problem 61 - Calculus

Question

Are the statements in Problems \(59-62\) true or false? Give an explanation for your answer. The differential equation \(d y / d x-x y=x\) can be solved by separation of variables.

Step-by-Step Solution

Answer

False, separation of variables is not applicable; it needs another method like integrating factor.

Step 1: Identify the Form of the Differential Equation

The given differential equation is \( \frac{dy}{dx} - x y = x \). This equation appears to be a first-order linear differential equation.

Step 2: Check if Separation of Variables is Applicable

Separation of variables is possible if the equation can be written in the form \( g(y) dy = h(x) dx \). The given equation is linear, and it cannot be rearranged into the separation of variables form, because the \(xy\) term couples the variables in a way that cannot be split.

Step 3: Determine Appropriate Solution Method

Since separation of variables does not apply, we identify that this is a linear differential equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \). It can be solved using an integrating factor or other methods suitable for linear differential equations.

Key Concepts

Separation of variables

Separation of variables is a method used to solve certain differential equations. This technique is applicable when the differential equation can be expressed in the form \( g(y) dy = h(x) dx \). Each side of the equation contains only one of the variables, allowing us to solve by integrating both sides separately.

• Start by trying to rearrange the equation to isolate terms involving \(y\) on one side and terms involving \(x\) on the other.
• Once separated, integrate both sides with respect to their respective variables.
• The result is a solution expressed in terms of \(y\) and \(x\).

In the original exercise, the equation \( \frac{dy}{dx} - xy = x \) cannot be separated into this form. The presence of an \(xy\) term prevents splitting the variables, making separation of variables inapplicable.

Integrating factor

An integrating factor is a powerful tool to solve linear first-order differential equations of the form \( \frac{dy}{dx} + P(x) y = Q(x) \). Here, the goal is to multiply the entire equation by a special function that simplifies it.

• The integrating factor is typically \( e^{\int P(x) \, dx} \) for a standard linear equation.
• Multiply through by this factor to make the left-hand side a perfect derivative: \( \frac{d}{dx} \left( \text{integrating factor} \times y \right) = \text{integrating factor} \times Q(x) \).
• Integrate both sides again to solve for \(y\).

Using this method transforms the original equation into a more manageable form, allowing us to find the solution effectively.

Differential equation

A differential equation is an equation that relates a function with its derivatives. Differential equations are used extensively to model real-world phenomena in engineering, physics, economics, and more. They are characterized by:

• Order: The highest derivative present in the equation.
• Linearity: Linear equations involve derivatives and the function itself in a linear manner.
• Type: Ordinary (ODE) if it involves derivatives of one variable, partial (PDE) if it involves multiple variables.

The differential equation in the exercise, \( \frac{dy}{dx} - xy = x \), is a first-order linear ordinary differential equation (ODE). This means it involves the first derivative of \(y\) with respect to \(x\), and all terms are either the first power of \(y\) or its derivatives.

Solution methods for linear differential equations

Linear differential equations come with systematic methods for finding solutions. For a first-order linear differential equation like \( \frac{dy}{dx} + P(x) y = Q(x) \), we typically use methods like the integrating factor, as described earlier.

• Integrating Factor Method: Multiply the equation by the integrating factor to make integration simpler.
• Variation of Parameters: A more generalized method allowing for non-constant coefficients, useful for second-order or higher linear ODEs.
• Laplace Transforms: Particularly helpful for linear differential equations with constant coefficients, transforming them to algebraic equations.

In the provided exercise, the integrating factor is the suitable choice because of the form of the equation. Applying this method reduces the complexity and leads us to the solution directly.