solve the differential equation y' + y = x*y^(2/3)

We are tasked with solving the first-order differential equation:

\[
y' + y = x y^{2/3}
\]

### Step 1: Simplifying the Equation
First, we rewrite the equation for clarity:

\[
\frac{dy}{dx} + y = x y^{2/3}
\]

This is not linear due to the \( y^{2/3} \) term, but we will attempt a substitution to simplify the equation.

### Step 2: Substitution
Let us use the substitution:

\[
y = u^3
\]

Then, \( y' = 3u^2 \frac{du}{dx} \). Substituting into the original equation gives:

\[
3u^2 \frac{du}{dx} + u^3 = x u^2
\]

### Step 3: Simplify the New Equation
We can divide the whole equation by \( u^2 \) (assuming \( u \neq 0 \)):

\[
3 \frac{du}{dx} + u = x
\]

This is now a linear first-order differential equation.

### Step 4: Solve the Linear Differential Equation
The equation \( 3 \frac{du}{dx} + u = x \) can be solved using an integrating factor.

The integrating factor is:

\[
\mu(x) = e^{\int \frac{1}{3} dx} = e^{x/3}
\]

Multiply both sides of the equation by \( e^{x/3} \):

\[
e^{x/3} \cdot 3 \frac{du}{dx} + e^{x/3} \cdot u = e^{x/3} \cdot x
\]

The left side becomes the derivative of \( 3u e^{x/3} \), so:

\[
\frac{d}{dx} \left( 3u e^{x/3} \right) = x e^{x/3}
\]

### Step 5: Integrate Both Sides
Now, integrate both sides with respect to \( x \):

\[
3u e^{x/3} = \int x e^{x/3} dx
\]

To solve this integral, we use integration by parts. Let \( v = x \) and \( dw = e^{x/3} dx \). After solving the integral, we get:

\[
\int x e^{x/3} dx = 3(x - 3)e^{x/3} + C
\]

Thus:

\[
3u e^{x/3} = 3(x - 3)e^{x/3} + C
\]

### Step 6: Solve for \( u \)
Divide through by \( 3e^{x/3} \):

\[
u = x - 3 + \frac{C}{3} e^{-x/3}
\]

### Step 7: Substitute Back for \( y \)
Recall that \( y = u^3 \), so:

\[
y = \left( x - 3 + \frac{C}{3} e^{-x/3} \right)^3
\]

This is the general solution to the given differential equation.

---

Let me know if you need more clarification on any steps!

### Related Questions:
1. How can we verify the solution to a differential equation?
2. What other methods can solve non-linear differential equations?
3. Can you solve the equation using a numerical method like Euler’s method?
4. How would the solution change if an initial condition were provided?
5. What is the role of the integrating factor in linear differential equations?

#### Tip:
When solving differential equations, substitutions often simplify non-linear terms, transforming them into solvable forms.

This article provides a step-by-step solution to the differential equation y' + y = x*y^(2/3). It covers important calculus concepts, including substitution and integrating factors, making it ideal for advanced high school or undergraduate students.

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