29. (a) Find the general solution of the differential equation $$2x^2 \frac{dy}{dx} = y^2 + 2xy.$$ (differential equation, general solution, calculus, homogeneous equation)

Step 1: Rearrange the differential equation

First, we rearrange the given differential equation to express $\frac{dy}{dx}$ in terms of $x$ and $y$. This helps us identify the type of differential equation we are dealing with. Dividing both sides by $2x^2$ gives us the standard form.

Step 2: Check for homogeneity and substitute

This is a homogeneous differential equation because all terms in the numerator and denominator have the same degree (degree 2). To solve it, we use the substitution $y = vx$, which implies $\frac{dy}{dx} = v + x\frac{dv}{dx}$. We substitute $y=vx$ into the equation and simplify.

Step 3: Separate variables

Now we substitute $v + x\frac{dv}{dx}$ for $\frac{dy}{dx}$ and rearrange the equation to separate the variables $v$ and $x$. We move $v$ to the right side, combine terms, and then isolate $dv$ and $dx$ on opposite sides of the equation.

Step 4: Integrate both sides

We integrate both sides of the separated equation. The integral of $v^{-2}$ is $-v^{-1}$, and the integral of $x^{-1}$ is $\ln|x|$. We add the constant of integration, $C$, on one side.

Step 5: Substitute back for $v$ and simplify

Finally, we substitute back $v = \frac{y}{x}$ into the integrated equation to express the solution in terms of $y$ and $x$. We can also rewrite the constant $-C$ as a new constant $C_1$ for simplicity and solve for $y$.

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