The given differential equation is:

$\frac{dy}{dx} + 2y = x$

This is a first-order linear differential equation. We can solve it using an integrating factor. Here's the step-by-step solution:

Step 1: Write the equation in standard form

The equation is already in the standard form of a first-order linear differential equation:

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

Where $P(x) = 2$ and $Q(x) = x$.

Step 2: Find the integrating factor

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

$\mu(x) = e^{\int P(x) \, dx} = e^{\int 2 \, dx} = e^{2x}$

Step 3: Multiply both sides of the equation by the integrating factor

Multiplying both sides of the differential equation by $e^{2x}$:

$e^{2x} \frac{dy}{dx} + 2e^{2x} y = x e^{2x}$

The left side is the derivative of $y e^{2x}$, so we can write:

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

Step 4: Integrate both sides

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

$y e^{2x} = \int x e^{2x} \, dx$

We can solve the integral on the right-hand side using integration by parts. Let:

• $u = x$, so $du = dx$
• $dv = e^{2x} dx$, so $v = \frac{1}{2} e^{2x}$

Now apply the integration by parts formula:

$\int x e^{2x} \, dx = \frac{1}{2} x e^{2x} - \frac{1}{2} \int e^{2x} dx$

$= \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + C$

Step 5: Solve for $y$

Now substitute this back into the equation:

$y e^{2x} = \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + C$

Divide through by $e^{2x}$:

$y = \frac{1}{2} x - \frac{1}{4} + C e^{-2x}$

Final Solution:

The general solution to the differential equation is:

$y(x) = \frac{1}{2} x - \frac{1}{4} + C e^{-2x}$

Where $C$ is the constant of integration.

Would you like more details on any part of this solution?

Related Questions:

1. How do you find the integrating factor for non-constant $P(x)$?
2. Can this method solve all first-order linear differential equations?
3. What is integration by parts and how does it work?
4. What is the physical interpretation of a first-order linear differential equation?
5. How does the general solution change if the equation has different initial conditions?

Tip:

Always check the standard form of a differential equation before applying the integrating factor method—it simplifies solving significantly!