We are given the equation:

$(2a^2 - r^2) \, dr = r^3 \sin\theta \, d\theta$ with the initial condition $\theta = 0$ and $r = a$.

We want to solve this equation or analyze it further. To do that, let's follow these steps:

Step 1: Simplify the given differential equation

Separate the variables $r$ and $\theta$ to make it easier to integrate.

$\frac{2a^2 - r^2}{r^3} \, dr = \sin\theta \, d\theta$

Step 2: Integrate both sides

We need to find the integrals of both sides.

Left-hand side (in terms of $r$):

$\int \frac{2a^2 - r^2}{r^3} \, dr = \int \left( \frac{2a^2}{r^3} - \frac{r^2}{r^3} \right) dr = \int \frac{2a^2}{r^3} \, dr - \int \frac{1}{r} \, dr$ First term: $\int \frac{2a^2}{r^3} \, dr = -\frac{a^2}{r^2}$ Second term: $\int \frac{1}{r} \, dr = \ln |r|$ So, the integral of the left-hand side is: $-\frac{a^2}{r^2} - \ln |r|$

Right-hand side (in terms of $\theta$):

$\int \sin\theta \, d\theta = -\cos\theta$

Step 3: Write the integrated equation

After integrating both sides, we get: $-\frac{a^2}{r^2} - \ln |r| = -\cos\theta + C$ where $C$ is the constant of integration.

Step 4: Apply the initial condition $\theta = 0$, $r = a$

Substitute $\theta = 0$ and $r = a$ into the equation: $-\frac{a^2}{a^2} - \ln |a| = -\cos(0) + C$ $-1 - \ln a = -1 + C$ This simplifies to: $C = -\ln a$

Step 5: Substitute the constant back into the equation

Now substitute $C = -\ln a$ back into the equation: $-\frac{a^2}{r^2} - \ln |r| = -\cos\theta - \ln a$ Simplify the equation: $-\frac{a^2}{r^2} - \ln \frac{r}{a} = -\cos\theta$ Or, equivalently: $\frac{a^2}{r^2} + \ln \frac{r}{a} = \cos\theta$

Final Equation:

$\frac{a^2}{r^2} + \ln \frac{r}{a} = \cos\theta$

This is the solution to the given differential equation.

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Would you like further details or have any questions?

Here are 5 related questions you could explore:

1. How does this equation change if we vary the initial condition for $\theta$?
2. What happens to the solution when $r$ approaches infinity?
3. How do we interpret the constants in physical problems involving differential equations?
4. Can you generalize this process for other trigonometric functions of $\theta$?
5. How would you solve this equation numerically if no exact solution were possible?

Tip: When integrating differential equations, always make sure to account for constants of integration properly—this is crucial for applying initial conditions later.