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The differential equation given is:

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

To solve this, we can divide both sides by \( x^2 \) (assuming \( x \neq 0 \)) to simplify:

\[
\frac{dy}{dx} = 1 + \frac{y}{x} + \frac{y^2}{x^2}
\]

This is a nonlinear first-order differential equation. To proceed, let’s use the substitution \( v = \frac{y}{x} \), where \( y = vx \). Differentiating \( y = vx \) with respect to \( x \) using the product rule gives:

\[
\frac{dy}{dx} = v + x \frac{dv}{dx}
\]

Substituting \( y = vx \) and \( \frac{dy}{dx} = v + x \frac{dv}{dx} \) into the equation, we get:

\[
v + x \frac{dv}{dx} = 1 + v + v^2
\]

Now, we can cancel \( v \) from both sides:

\[
x \frac{dv}{dx} = 1 + v^2
\]

This is a separable differential equation in terms of \( v \) and \( x \). Rewriting, we have:

\[
\frac{dv}{1 + v^2} = \frac{dx}{x}
\]

Integrating both sides:

\[
\int \frac{dv}{1 + v^2} = \int \frac{dx}{x}
\]

The integral on the left side is \( \arctan(v) \), and the integral on the right side is \( \ln|x| \), so:

\[
\arctan(v) = \ln|x| + C
\]

Substituting back \( v = \frac{y}{x} \), we get:

\[
\arctan\left(\frac{y}{x}\right) = \ln|x| + C
\]

This is the implicit solution to the differential equation. 

Would you like a further breakdown of each step or have any questions?

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Here are five related questions for further exploration:

1. How would we solve if the equation was linear instead?
2. What is an exact differential equation, and how does it differ from this one?
3. How does the method of substitution help simplify nonlinear differential equations?
4. What would the solution look like if \( y(1) = 0 \) as an initial condition?
5. How does the arctangent function affect the solution’s behavior at different values of \( x \)?

**Tip:** In differential equations, substitution is a powerful tool to reduce complex equations to simpler forms, making them easier to integrate.

The solution of x^2 dy/dx = x^2 + xy + y^2

Explore the step-by-step solution to the nonlinear differential equation x^2 dy/dx = x^2 + xy + y^2 using substitution. Learn key mathematical concepts such as nonlinear differential equations, separable equations, and the substitution method. Suitable for undergraduate mathematics students.

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