Question

Put each of the following first-order linear differential equations into standard form. Identify \(p(x)\) and \(q(x)\) for each equation. a. \(y^{\prime}=3 x-4 y\) b. \(\frac{3 x y^{\prime}}{4 y-3}=2(\) here \(x>0)\) c. \(y=3 y^{\prime}-4 x^{2}+5\)

Short answer

a: \(p(x)=4, q(x)=3x\); b: \(p(x)=-\frac{8}{3x}, q(x)=-\frac{6}{3x}\); c: \(p(x)=-\frac{1}{3}, q(x)=\frac{4x^2}{3}-\frac{5}{3}\).

Step-by-step solution

Step 1a: Rearrange the Equation

Given the equation is \(y' = 3x - 4y\). To transform this into the standard form of a first-order linear differential equation \(y' + p(x)y = q(x)\), we need to rearrange it. Add \(4y\) to both sides to give: \(y' + 4y = 3x\).

Step 1b: Identify p(x) and q(x)

The equation \(y' + 4y = 3x\) is now in standard form. Thus, we identify \(p(x) = 4\) and \(q(x) = 3x\).

Step 2a: Clear the Fraction

Given the equation is \(\frac{3 xy'}{4y-3} = 2\). Multiply both sides by \(4y-3\) to clear the fraction: \(3xy' = 2(4y-3)\).

Step 2b: Rearrange and Simplify

Expand the right-hand side: \(3xy' = 8y - 6\). Now, divide through by \(3x\) to isolate \(y'\): \(y' = \frac{8y - 6}{3x}\). Rearrange to standard form \(y' - \frac{8}{3x}y = -\frac{6}{3x}\).

Step 2c: Identify p(x) and q(x)

The equation \(y' - \frac{8}{3x}y = -\frac{6}{3x}\) is in standard form. Identify \(p(x) = -\frac{8}{3x}\) and \(q(x) = -\frac{6}{3x}\).

Step 3a: Express as y' Term

Given the equation \(y = 3y' - 4x^2 + 5\). Rearrange to solve for \(y'\): \(3y' = y + 4x^2 - 5\). Divide through by 3: \(y' = \frac{y}{3} + \frac{4x^2}{3} - \frac{5}{3}\).

Step 3b: Rearrange to Standard Form

Rearrange the equation \(y' = \frac{y}{3} + \frac{4x^2}{3} - \frac{5}{3}\) to standard form: \(y' - \frac{1}{3}y = \frac{4x^2}{3} - \frac{5}{3}\).

Step 3c: Identify p(x) and q(x)

In standard form \(y' - \frac{1}{3}y = \frac{4x^2}{3} - \frac{5}{3}\), identify \(p(x) = -\frac{1}{3}\) and \(q(x) = \frac{4x^2}{3} - \frac{5}{3}\).

Key concepts

Standard Form

When dealing with first-order linear differential equations, it's often helpful to express the equation in a specific structure known as the *standard form*. This form looks like this: \(y' + p(x)y = q(x)\). The equation must be rearranged to fit this structure since it greatly simplifies the process of solving the differential equation.

The standard form separates the derivative term \(y'\) and the term involving \(y\), labeled as \(p(x)\), on one side of the equation while isolating the function \(q(x)\), on the other side. - **Consistency:** This form provides a consistent way to identify and extract key components \(p(x)\) and \(q(x)\). - **Solving Strategy:** Having it in this form allows you to apply solution methods like the integrating factor method easily.

In practice, transforming any given differential equation into this standard form simplifies our task significantly.

p(x) and q(x) Identification

Once a differential equation has been rearranged into the standard form \(y' + p(x)y = q(x)\), the next step is to identify the functions \(p(x)\) and \(q(x)\).

- **\(p(x)\):** This function is the coefficient of \(y\) in the standard form equation. It represents the part of the equation that modifies the \(y\) variable directly. - **\(q(x)\):** This function is isolated on the right side of the equation and often denotes what the derivative balances out, or the way \(y'\) is being driven by other factors.

### ExampleConsider the equation that has been rewritten as \(y' + 4y = 3x\) from step 1b:- Here, \(p(x) = 4\) shows how \(y\) is influenced directly in the differential equation.- And, \(q(x) = 3x\) reflects what \(y'\) compensates in this dynamic system.Identifying \(p(x)\) and \(q(x)\) is crucial as it allows one to understand the interplay between the unknown function and its derivatives.

Rearranging Equations

Rearranging equations is a key skill when dealing with differential equations, ensuring they are suitable for identifying key components or applying solution techniques. Transforming an equation into standard form often involves strategic manipulation of terms.

### Steps in Rearranging:1. **Move Terms Appropriately:** When given an equation like \(y' = 3x - 4y\), add \(4y\) to both sides to help structure it in a recognizable form: \(y' + 4y = 3x\).2. **Clear Fractions:** If fractions are present, such as in the equation \(\frac{3xy'}{4y-3} = 2\), multiplying through by \(4y-3\) removes the fraction and places the equation in a more manageable form for further rearrangement.3. **Isolate y' and y Terms:** Ensure that \(y'\) and any terms involving \(y\) are on one side, solidifying the equation in the standard form to easily identify \(p(x)\) and \(q(x)\).

Rearranging not only transforms these equations into the standard form but also paves the way for a streamline solution process.