The given first-order differential equation is:

$y' = xy + 3y - 2x - 6$

Step 1: Express in Standard Form

Rewriting it in the form:

$\frac{dy}{dx} - xy - 3y = -2x - 6$

This is a first-order linear differential equation of the form:

$\frac{dy}{dx} + P(x)y = Q(x)$

where:

• $P(x) = -(x + 3)$
• $Q(x) = -2x - 6$

Step 2: Find the Integrating Factor (IF)

The integrating factor is given by:

$IF = e^{\int P(x) dx} = e^{\int -(x+3) dx}$

Computing the integral:

$\int -(x+3) dx = -\frac{x^2}{2} - 3x$

Thus, the integrating factor is:

$IF = e^{-\frac{x^2}{2} - 3x}$

Step 3: Multiply the Differential Equation by the Integrating Factor

Multiplying everything by $e^{-\frac{x^2}{2} - 3x}$, we get:

$e^{-\frac{x^2}{2} - 3x} \frac{dy}{dx} - (x+3)e^{-\frac{x^2}{2} - 3x}y = (-2x - 6)e^{-\frac{x^2}{2} - 3x}$

Since the left-hand side is the derivative of $y$ multiplied by the integrating factor, it simplifies to:

$\frac{d}{dx} \left( y e^{-\frac{x^2}{2} - 3x} \right) = (-2x - 6)e^{-\frac{x^2}{2} - 3x}$

Step 4: Integrate Both Sides

Integrating both sides:

$\int d\left( y e^{-\frac{x^2}{2} - 3x} \right) = \int (-2x - 6)e^{-\frac{x^2}{2} - 3x} dx$

Using integration by parts or substitution techniques, solving the right-hand side gives:

$y e^{-\frac{x^2}{2} - 3x} = C + \int (-2x - 6)e^{-\frac{x^2}{2} - 3x} dx$

Step 5: Solve for $y(x)$

$y = e^{\frac{x^2}{2} + 3x} \left( C + \int (-2x - 6)e^{-\frac{x^2}{2} - 3x} dx \right)$

This expression represents the general solution of the given differential equation.

Would you like me to further evaluate the integral explicitly? Let me know how detailed you need the solution.

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Related Questions:

1. How do you determine the integrating factor for first-order linear differential equations?
2. What are the steps to solve first-order linear differential equations?
3. How do you apply integration by parts in differential equations?
4. What are some real-world applications of first-order linear differential equations?
5. How do you solve non-homogeneous differential equations?

Tip:

Always check whether a differential equation is separable before proceeding with other methods. Separable equations are often easier to solve.