The equation shown in the image involves a differential equation with an elliptic integral form. The equation is:

$\frac{dr}{\sqrt{\frac{1}{r} - \frac{1}{r_0}}} = \sqrt{2Gm} \, dt$

To solve this differential equation for $r(t)$, we need to perform a few steps.

Step 1: Rewrite the equation in a more manageable form

The equation can be simplified by making a substitution. Let:

$u = \frac{1}{r}$

This gives us:

$dr = -\frac{1}{u^2} \, du$

Now, substituting this into the differential equation, we get:

$-\frac{1}{u^2} \, \frac{du}{\sqrt{u - \frac{1}{r_0}}} = \sqrt{2Gm} \, dt$

Step 2: Separation of variables

Next, we'll try to separate the variables $u$ and $t$ for integration:

$-\frac{du}{u^2 \sqrt{u - \frac{1}{r_0}}} = \sqrt{2Gm} \, dt$

Step 3: Integration

The integral on the left-hand side is an elliptic integral, and it requires special methods to solve. You can recognize this as a standard form that involves an elliptic integral of the first kind. Symbolically, we write the solution as:

$F(u) = \int \frac{du}{u^2 \sqrt{u - \frac{1}{r_0}}}$

Solving this exactly will result in a complex expression involving elliptic functions. For practical purposes, this can be solved numerically or expressed in terms of known elliptic integrals.

Step 4: Solving for $r(t)$

After solving the elliptic integral, we would invert the substitution $u = \frac{1}{r}$ to find $r(t)$. The final solution will depend on the initial conditions and how the integral evaluates.

Would you like a more detailed step-by-step guide on performing the integral, or would you like help with a numerical approach to finding $r(t)$?

Here are five related questions you may find helpful:

1. What are elliptic integrals, and how are they typically solved?
2. What is the physical context of this differential equation (e.g., orbital mechanics)?
3. How can this equation be solved numerically using methods such as Runge-Kutta?
4. How does the substitution $u = \frac{1}{r}$ simplify solving problems in gravitational systems?
5. What are the differences between elliptic integrals of the first, second, and third kind?

Tip: Elliptic integrals often arise in problems involving conic sections (like ellipses) and gravitational systems. Numerical solutions can often provide more insights than exact ones in such cases.