Question

True or False? In Exercises \(81-83,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The differential equation \(y^{\prime}=x y-2 y+x-2\) can be written in separated variables form.

Short answer

False. The given differential equation \(y^{\prime}=x y-2 y+x-2\) cannot be written in separated variables form.

Step-by-step solution

Examine the given equation

The given differential equation is \(y^{\prime}=x y-2 y+x-2\). We attempt to separate the variables by expressing this in the form \(dy/dx = f(y)g(x)\).

Group similar terms

Rearrange like-terms together by grouping all terms with \(y\) and those without \(y\): this results in \(y^{\prime}= y(x-2) + x - 2\).

Attempt to separate variables

In an attempt to separate the variables \(x\) and \(y\), we see it's not possible to express the equation in the form \(dy/dx = f(y)g(x)\).

Key concepts

Separable Equations

Separable equations are a specific type of differential equations that can be written in a form that allows us to separate the variables into two sides of the equation. This means each side of the equation contains only one variable. Typically, separable differential equations look like this: \( \frac{dy}{dx} = f(y)g(x) \).

The power of this form is that it allows us to simplify the problem by integrating both sides independently. When variables can be easily separated, solving the differential equation becomes more manageable.

• Place all terms involving \(y\) on one side with \(dy\).
• Place all terms involving \(x\) on the other side with \(dx\).
• Integrate both sides to find the solution.

However, not all equations are separable. If the terms involving the variables are too intertwined, like in the equation we are examining, separation may not be feasible.

Variable Separation

Variable separation is a technique used to solve separable differential equations by disentangling the variables. This approach simplifies the integration process, leading us to the solution of the differential equation.

To perform variable separation:

• Identify if the equation is separable by checking if it can be expressed as \( \frac{dy}{dx} = h(y)k(x) \).
• Arrange all \(y\)-dependent terms with \(dy\) on one side: \( h(y) \ dy \).
• Arrange all \(x\)-dependent terms with \(dx\) on the other side: \( k(x) \ dx \).
• Integrate both sides independently to solve for \(y\) in terms of \(x\).

In some cases, this isn't straightforward or possible. If an equation, like our example, cannot be rewritten to only involve either \(y\) or \(x\) on each respective side, the equation isn't naturally separable, and other methods might be needed.

Mathematical Proof

A mathematical proof is essential in demonstrating whether a mathematical statement is true or false. To determine if an equation like the one in our example can be separated, we conduct a mathematical proof by trying to transform it into a separable form.

This involves:

• Attempting to rearrange the equation to isolate terms involving \(y\) and \(x\).
• Checking if the equation aligns with the form \( \frac{dy}{dx} = f(y)g(x) \).
• If successful, variable separation is possible, proving the statement true.
• If unsuccessful, providing logical or algebraic reasoning to show why separation cannot occur.

For our provided equation, finding that it can't be transformed into a separable form is part of the proof that the statement is false. Proving statements in mathematics often involves showing all possible configurations, like in this case, where separation was attempted through rearrangement.

Equation Rearrangement

Rearrangement in differential equations is a tactical operation where we organize terms to try to make them more workable or to fit a certain form, like separable equations. It's a preliminary step to check if variable separation is applicable.

To rearrange a differential equation effectively:

• Identify and group similar terms. Separate components of the equation into parts that involve \(y\), \(x\), or neither.
• Simplify where possible, to bring terms into a format that will aid in further processing, like separating or factoring terms.
• Check if the rearrangement leads the equation to reveal a hidden structure, such as a product of a function of \(y\) and a function of \(x\).

The differential equation \(y^{\prime} = x y - 2 y + x - 2\) can be grouped as \(y(x-2) + (x-2)\), but ultimately, further direct separation isn't possible, illustrating that rearrangement can reveal, but not always realize, simplifiable forms.