The given differential equation is:

$y'(x) - 3y(x) = x$

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

$y'(x) + p(x)y(x) = q(x)$

where $p(x) = -3$ and $q(x) = x$.

Step 1: Find the Integrating Factor (IF)

The integrating factor, $\mu(x)$, is given by:

$\mu(x) = e^{\int p(x) \, dx}$

Since $p(x) = -3$, we compute the integral:

$\int -3 \, dx = -3x$

So, the integrating factor becomes:

$\mu(x) = e^{-3x}$

Step 2: Multiply the Equation by the Integrating Factor

Now, multiply both sides of the differential equation by $e^{-3x}$:

$e^{-3x} y'(x) - 3e^{-3x} y(x) = x e^{-3x}$

This simplifies to:

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

Step 3: Integrate Both Sides

Next, integrate both sides with respect to $x$:

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

The left-hand side is simply:

$e^{-3x} y(x)$

To integrate $x e^{-3x}$, we use integration by parts. Let:

• $u = x$ and $dv = e^{-3x} dx$
• Then $du = dx$ and $v = \frac{e^{-3x}}{-3}$

Using integration by parts:

$\int x e^{-3x} \, dx = \frac{x e^{-3x}}{-3} - \int \frac{e^{-3x}}{-3} \, dx$

$= \frac{-x e^{-3x}}{3} + \frac{e^{-3x}}{9}$

Thus, the integral becomes:

$e^{-3x} y(x) = \frac{-x e^{-3x}}{3} + \frac{e^{-3x}}{9} + C$

where $C$ is the constant of integration.

Step 4: Solve for $y(x)$

Now, multiply both sides by $e^{3x}$ to solve for $y(x)$:

$y(x) = \frac{-x}{3} + \frac{1}{9} + C e^{3x}$

Final Solution

The general solution to the differential equation is:

$y(x) = \frac{-x}{3} + \frac{1}{9} + C e^{3x}$

where $C$ is an arbitrary constant.

---

Would you like further details on any of the steps or explanations?

Here are 5 related questions you might explore next:

1. What is the general approach to solving other first-order linear differential equations?
2. How do you apply integration by parts in this type of problem?
3. What are the different methods to solve linear differential equations with non-constant coefficients?
4. How does the integrating factor method work for equations with other forms of $p(x)$?
5. How can you use this solution to find particular solutions if initial conditions are given?

Tip: Always check for an appropriate integrating factor, as it simplifies many linear differential equations.