The given differential equation is:

$t x' = x + 2\sqrt{tx}$

Where $x'$ denotes the derivative of $x$ with respect to $t$.

Let's work through solving this step by step.

Step 1: Rewrite the equation

First, divide through by $t$ to isolate $x'$ on the left-hand side:

$x' = \frac{x}{t} + \frac{2\sqrt{tx}}{t}$

Simplify the second term:

$x' = \frac{x}{t} + \frac{2}{\sqrt{t}} \cdot \sqrt{\frac{x}{t}}$

Step 2: Substitution

Let’s introduce a substitution to simplify the equation. Set $v(t) = \sqrt{x(t)/t}$, which means:

$x(t) = v(t)^2 t$

Now differentiate $x(t)$ with respect to $t$:

$x' = 2v(t) v'(t) t + v(t)^2$

Substitute this expression for $x'$ back into the original equation:

$2v(t) v'(t) t + v(t)^2 = v(t)^2 t + 2v(t)$

Step 3: Simplify and solve

Now, simplify the equation:

$2v(t) v'(t) t = 2v(t)$

Cancel $2v(t)$ from both sides (assuming $v(t) \neq 0$):

$v'(t) t = 1$

This is a separable differential equation. Solve for $v(t)$:

$v'(t) = \frac{1}{t}$

Integrating both sides with respect to $t$:

$v(t) = \ln|t| + C$

Step 4: Back-substitute for $x(t)$

Recall that $v(t) = \sqrt{x(t)/t}$, so:

$\sqrt{\frac{x(t)}{t}} = \ln|t| + C$

Square both sides:

$\frac{x(t)}{t} = (\ln|t| + C)^2$

Finally, multiply by $t$:

$x(t) = t (\ln|t| + C)^2$

Final Solution:

The general solution to the differential equation is:

$x(t) = t (\ln|t| + C)^2$

Would you like further clarification or details on any step? Here are five related questions to explore the topic further:

1. How do you solve separable differential equations?
2. What other substitution techniques are commonly used in solving differential equations?
3. How does the method of integrating factors apply to first-order linear differential equations?
4. Can the solution to this differential equation be used for specific initial conditions?
5. How does one handle boundary conditions in non-linear differential equations?

Tip: Substitution is a powerful method for simplifying complex differential equations—always look for opportunities to reduce terms to familiar forms.