 Solve the differential equation 𝑑𝑦
𝑑𝑥
+
𝑦
𝑥
= x
3

The given differential equation is:

\[
\frac{dy}{dx} + \frac{y}{x} = x^3
\]

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

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

where \( P(x) = \frac{1}{x} \) and \( Q(x) = x^3 \).

### Step 1: Find the Integrating Factor (IF)

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

\[
\mu(x) = e^{\int P(x) dx} = e^{\int \frac{1}{x} dx} = e^{\ln|x|} = |x|
\]

So, the integrating factor is \( \mu(x) = x \) (we can drop the absolute value if we assume \( x > 0 \)).

### Step 2: Multiply the whole equation by the integrating factor

Multiplying both sides of the differential equation by \( x \):

\[
x \frac{dy}{dx} + y = x^4
\]

### Step 3: Solve the equation

The left-hand side of the equation is now the derivative of \( yx \):

\[
\frac{d}{dx}(yx) = x^4
\]

Integrating both sides with respect to \( x \):

\[
yx = \int x^4 dx = \frac{x^5}{5} + C
\]

where \( C \) is the constant of integration.

### Step 4: Solve for \( y \)

Finally, divide by \( x \) to isolate \( y \):

\[
y = \frac{x^4}{5} + \frac{C}{x}
\]

Thus, the solution to the differential equation is:

\[
y = \frac{x^4}{5} + \frac{C}{x}
\]

### Do you want any further details or have any questions?

Here are 5 questions related to this topic:
1. What are some examples of first-order linear differential equations?
2. How is an integrating factor used in solving linear differential equations?
3. How do we solve differential equations with non-constant coefficients?
4. What is the significance of the constant of integration?
5. Can a linear differential equation always be solved using an integrating factor?

**Tip:** Always check if a first-order linear differential equation can be simplified using an integrating factor. It makes solving such equations straightforward.

Solve the differential equation dy/dx + y/x = x^3

Explore how to solve the first-order linear differential equation dy/dx + y/x = x^3 using the integrating factor method. This step-by-step solution covers key concepts in differential equations, including the use of integrating factors and the significance of constants of integration. Suitable for advanced high school students and undergraduates.

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