We are tasked with solving the differential equation:

$\frac{dx}{dt} = \frac{4x^2}{t^2} + \frac{x}{t} + 1$ using the substitution $z = \frac{2x}{t}$.

Step 1: Express $x$ in terms of $z$

The substitution $z = \frac{2x}{t}$ gives: $x = \frac{zt}{2}$

Step 2: Differentiate $x$ with respect to $t$

Now, differentiate $x = \frac{zt}{2}$ with respect to $t$ to get $\frac{dx}{dt}$. Using the product rule: $\frac{dx}{dt} = \frac{1}{2}\left( \frac{dz}{dt} t + z \right)$

Step 3: Substitute into the original equation

Now, substitute $x = \frac{zt}{2}$ and $\frac{dx}{dt} = \frac{1}{2} \left( \frac{dz}{dt} t + z \right)$ into the original equation: $\frac{1}{2} \left( \frac{dz}{dt} t + z \right) = \frac{4\left( \frac{zt}{2} \right)^2}{t^2} + \frac{\frac{zt}{2}}{t} + 1$

Simplifying the right-hand side: $\frac{4\left( \frac{zt}{2} \right)^2}{t^2} = \frac{4z^2 t^2}{4t^2} = z^2$ $\frac{\frac{zt}{2}}{t} = \frac{z}{2}$

Thus, the equation becomes: $\frac{1}{2} \left( \frac{dz}{dt} t + z \right) = z^2 + \frac{z}{2} + 1$

Step 4: Multiply through by 2

Multiply both sides of the equation by 2 to eliminate the fraction: $\frac{dz}{dt} t + z = 2z^2 + z + 2$

Step 5: Simplify the equation

Now, simplify the equation: $\frac{dz}{dt} t = 2z^2 + 2$ or $\frac{dz}{dt} = \frac{2z^2 + 2}{t}$

Step 6: Separate variables

Rearrange this to separate the variables: $\frac{dz}{2z^2 + 2} = \frac{dt}{t}$

Factor the left-hand side: $\frac{dz}{2(z^2 + 1)} = \frac{dt}{t}$

Step 7: Integrate both sides

Now, integrate both sides: $\int \frac{dz}{2(z^2 + 1)} = \int \frac{dt}{t}$

The integral on the left side is a standard arctangent integral, and the right side is a logarithmic integral: $\frac{1}{2} \int \frac{dz}{z^2 + 1} = \ln|t| + C$ $\frac{1}{2} \tan^{-1}(z) = \ln|t| + C$

Step 8: Solve for $z$

Now solve for $z$: $\tan^{-1}(z) = 2(\ln|t| + C)$ $z = \tan\left( 2(\ln|t| + C) \right)$

Step 9: Express $x$ in terms of $t$

Recall that $z = \frac{2x}{t}$, so: $\frac{2x}{t} = \tan\left( 2(\ln|t| + C) \right)$ $x = \frac{t}{2} \tan\left( 2(\ln|t| + C) \right)$

This is the general solution to the differential equation.

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Would you like further details or explanations on any part of this solution?

Here are 5 related questions for deeper exploration:

1. What is the physical interpretation of the substitution $z = \frac{2x}{t}$ in the context of the differential equation?
2. How do we interpret the constant $C$ in the solution, and what boundary conditions might be used to determine it?
3. How would you solve a similar differential equation where the right-hand side involves higher powers of $x$ or other functions of $x$?
4. Can you explain how the technique of separation of variables works in general for differential equations?
5. How would you modify this solution if the equation had a non-homogeneous term, such as $\frac{dx}{dt} = \text{some function of } x$?

Tip: Always try to recognize patterns in the equation, such as separable forms or simple substitutions, that allow you to simplify the equation and make it easier to solve.