 Solve using homogeneous differential equations for 3xy^2(dy/dx) =x^3+y^3

### Solution using **Homogeneous Differential Equations**:

We are given:  
\[
3x y^2 \frac{dy}{dx} = x^3 + y^3
\]

### Step 1: Check for Homogeneity
A differential equation is homogeneous if both \(M(x, y)\) and \(N(x, y)\) (in the form \(M(x, y) + N(x, y) \frac{dy}{dx} = 0\)) have the same degree. 

Let’s rewrite the given equation as:
\[
\frac{dy}{dx} = \frac{x^3 + y^3}{3x y^2}
\]

Notice that the degree of both the numerator \(x^3 + y^3\) and the denominator \(3x y^2\) is 3. Thus, this is a **homogeneous differential equation**.

---

### Step 2: Substitution \(y = vx\)  
We use the substitution \(y = vx \Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx}\).

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### Step 3: Substitute into the Differential Equation  
Now, substitute \(y = vx\) and \(\frac{dy}{dx} = v + x \frac{dv}{dx}\) into:
\[
\frac{dy}{dx} = \frac{x^3 + (vx)^3}{3x (vx)^2}
\]

Simplify:
\[
\frac{dy}{dx} = \frac{x^3 + v^3 x^3}{3x^3 v^2} = \frac{(1 + v^3) x^3}{3v^2 x^3} = \frac{1 + v^3}{3v^2}
\]

Thus, the equation becomes:
\[
v + x \frac{dv}{dx} = \frac{1 + v^3}{3v^2}
\]

---

### Step 4: Rearranging the Equation  
Rearrange to isolate terms involving \(v\) and \(x\):
\[
x \frac{dv}{dx} = \frac{1 + v^3}{3v^2} - v
\]

Simplify:
\[
x \frac{dv}{dx} = \frac{1 + v^3 - 3v^3}{3v^2} = \frac{1 - 2v^3}{3v^2}
\]

Thus, the equation becomes:
\[
\frac{dv}{dx} = \frac{1 - 2v^3}{3v^2 x}
\]

---

### Step 5: Separate the Variables  
Separate the variables \(v\) and \(x\):
\[
3v^2 \, dv = \frac{1 - 2v^3}{x} \, dx
\]

---

### Step 6: Integrate Both Sides  
Integrate both sides:

For the left side:
\[
\int 3v^2 \, dv = \int \left(1 - 2v^3\right) \frac{dx}{x}
\]

The integral of \(3v^2 \, dv\) is:
\[
v^3 + C_1
\]

For the right side:
\[
\int \left(1 - 2v^3\right) \frac{dx}{x} = \ln|x| - 2v^3 \ln|x| + C_2
\]

---

### Step 7: Combine the Results  
We now have:
\[
v^3 = \ln|x| - 2v^3 \ln|x| + C
\]

Replace \(v = \frac{y}{x}\) to express the solution in terms of \(x\) and \(y\). You can further simplify or leave this as an implicit solution.

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### Related Questions  
1. How do you verify if a differential equation is homogeneous?  
2. Can we always solve homogeneous equations by the substitution \(y = vx\)?  
3. How do you solve non-homogeneous differential equations?  
4. What is the role of integrating factors in differential equations?  
5. How do exact differential equations differ from homogeneous ones?

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### Tip  
When using substitutions in homogeneous equations, always be careful with algebraic simplifications after substitution—small mistakes can propagate through the problem!

Solve using homogeneous differential equations for 3xy^2(dy/dx) = x^3 + y^3

This detailed solution explains how to solve the homogeneous differential equation 3xy^2(dy/dx) = x^3 + y^3. Learn step-by-step methods using substitution, integration, and algebraic simplification, suitable for advanced high school or college-level students.

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