Question

Put the equation \(\frac{(x+3) y^{\prime}}{2 x-3 y-4}=5\) into standard form and identify \(p(x)\) and \(q(x)\).

Short answer

The standard form is \(y^{\prime} + \frac{15}{x+3} y = \frac{10x - 20}{x+3}\), with \(p(x) = \frac{15}{x+3}\) and \(q(x) = \frac{10x - 20}{x+3}\).

Step-by-step solution

Recognize the Equation Form

The given equation is \( \frac{(x+3) y^{\prime}}{2 x-3 y-4}=5 \). To convert this into standard form, identify that it is a linear differential equation of the form \( \frac{d y}{d x} + P(x) y = Q(x) \).

Multiply to Clear Denominator

Multiply both sides of the equation by \(2x - 3y - 4\) to clear the denominator: \[(x+3) y^{\prime} = 5(2x - 3y - 4)\].

Expand the Right Side

Expand the right side of the equation: \[(x+3) y^{\prime} = 10x - 15y - 20\].

Rearrange terms

Bring all terms involving \(y\) to the left side and other terms to the right side:\[(x+3) y^{\prime} + 15y = 10x - 20\].

Solve for \(y'\) Coefficient

Divide the entire equation by \(x+3\) to isolate \(y^{\prime}\):\[y^{\prime} + \frac{15}{x+3} y = \frac{10x - 20}{x+3}\].

Identify \(p(x)\) and \(q(x)\)

From the equation \(y^{\prime} + \frac{15}{x+3} y = \frac{10x - 20}{x+3}\), identify \(p(x) = \frac{15}{x+3}\) and \(q(x) = \frac{10x - 20}{x+3}\).

Key concepts

Standard Form

To convert a differential equation into standard form, we aim to express it as \[ y' + P(x) y = Q(x) \]. This format is crucial for solving the equation using various techniques, such as finding integrating factors. In this form, \( y' \) represents the derivative of the function \( y \) with respect to \( x \), while \( P(x) \) and \( Q(x) \) are continuous functions that may involve \( x \) but not \( y \).

To reach this form from an existing equation, manipulation is necessary. Start by clearing any fractions by multiplying both sides by the denominator. Then, ensure all terms involving \( y \) are on one side of the equation, and all other terms are on the other side. This setup makes it straightforward to identify each component of the equation for further analysis.

p(x) and q(x) Identification

After transforming the equation into its standard form, the next step is identifying \( p(x) \) and \( q(x) \). These functions are critical because they directly impact the behavior and solutions of the differential equation.

In our problem, once the equation was rendered into the format \[ y' + \frac{15}{x+3} y = \frac{10x - 20}{x+3} \], it became straightforward to see that \( p(x) = \frac{15}{x+3} \) and \( q(x) = \frac{10x - 20}{x+3} \).

Identifying these functions involves isolating the coefficient of \( y \) (which is \( p(x) \)) from the non-homogeneous part (\( q(x) \)). Recognizing these parameters correctly is fundamental, as they will often determine the strategy for finding the solution to the differential equation.

Denominator Clearing

Clearing the denominator is a pivotal step in simplifying a differential equation. This process involves multiplying through by the denominator of any fraction present in the equation. Doing so removes the fraction and simplifies the equation into a polynomial form.

In the exercise, the original expression \( \frac{(x+3) y'}{2 x-3 y-4} = 5 \) can be cumbersome due to the fraction. Multiplying both sides by \( 2x - 3y - 4 \) removes the denominator, resulting in \((x+3) y' = 5(2x - 3y - 4)\). This step eases the manipulation of the equation and sets the stage to bring it to standard form. Removing denominators simplifies equations allowing you to work with integer coefficients, which are generally easier to manage.

Coefficients in Differential Equations

In differential equations, coefficients are the multiplied terms accompanying \( y \) and its derivatives. Understanding their role is vital, as they influence the form and the difficulty of solving the equation. Coefficients can typically vary with respect to \( x \) or be constants.

Within this exercise, after manipulating the equation, the coefficient of \( y' \) was adjusted by organizing it into standard form as \( y' + \frac{15}{x+3} y = \frac{10x - 20}{x+3} \). Here, \( \frac{15}{x+3} \) is the coefficient impacting \( y \), which associates with \( p(x) \).

Understanding the nature of such coefficients helps you anticipate the complexity of solutions and the algebra needed to reach those solutions. By identifying and working with these coefficients, derivations and solutions become more systematic and precise.