Question

Write the following first-order differential equations in standard form. $$ -x y^{\prime}=(3 x+2) y+x e^{x} $$

Short answer

The standard form is \(y' + \left(3 + \frac{2}{x}\right)y = -e^x\).

Step-by-step solution

Identify Standard Form

The standard form of a first-order differential equation is \( y' + P(x)y = Q(x) \). We need to manipulate the given equation to match this format.

Simplify the Given Equation

Start with the given equation: \[-xy' = (3x + 2)y + xe^x\]Divide every term by \(-x\) to get \[y' = -\left(\frac{3x + 2}{x}\right)y - e^x\].

Simplify the Right Side Further

Simplify \(-\left(\frac{3x + 2}{x}\right)y - e^x\) to:\[-\left(3 + \frac{2}{x}\right)y - e^x\]. Now we have:\[y' + \left(3 + \frac{2}{x}\right)y = -e^x\].

Write in Standard Form

The equation is now in its standard form:\[y' + P(x)y = Q(x)\],where \(P(x) = 3 + \frac{2}{x}\) and \(Q(x) = -e^x\).

Key concepts

Standard Form

The standard form of a first-order differential equation simplifies the process of solving and analyzing these types of equations. It is expressed as \( y' + P(x)y = Q(x) \). Here, \( y' \) represents the derivative of \( y \) with respect to \( x \), \( P(x) \) is a function of \( x \) that multiplies \( y \), and \( Q(x) \) is another function of \( x \).

By expressing the differential equation in this particular format, we can apply well-established methods to find the solution. This form helps us easily identify the terms that need attention for further manipulation. For our specific exercise, the given equation was not initially in standard form, necessitating additional steps to transform it.

• Format Uniformity: Facilitates consistency in solving.
• Clarity: Breaks down complex relationships between variables.
• Methodical Approach: Forms the basis for applying integration techniques.

Manipulation of Equations

Manipulating equations involves applying mathematical operations to rearrange or transform it into a desired format. Here, the manipulation process started with the original equation \(-xy' = (3x + 2)y + xe^x \).

To get the equation into standard form, we first needed to isolate the derivative \( y' \) on one side of the equation. By dividing every term by \(-x\), the expression is simplified step by step. This operation ensures that \( y' \) stands alone on one side, allowing us to shape the equation in standard form.

• Isolation of Terms: The goal is to isolate \( y' \) efficiently.
• Balance Operations: Ensure each step maintains the equation's balance.
• Strategic Division: Used here to manage and reduce coefficient complexity.

Simplification of Expressions

Simplifying expressions is an essential step in solving differential equations. In this exercise, the simplification of \(-\left(\frac{3x + 2}{x}\right)y - e^x \) involved breaking down the fraction \( \frac{3x + 2}{x} \) into its components.

This can be expressed as \(-(3 + \frac{2}{x})y - e^x \), which makes the equation clearer and more manageable. Simplification helps in not only achieving standard form but also in revealing the inherent relationships between different components of the equation.

• Component Decomposition: Simplifying complex fractions into single terms.
• Clarity Achievement: Aims to make each part of the equation understandable.
• Ease of Application: Facilitates easier identification of \( P(x) \) and \( Q(x) \).