We are given the following initial-value problem:

$\frac{dy}{dx} = \frac{x^2}{\sqrt{1 - y^2}}, \quad y(0) = 0$

Step 1: Separate the variables

We can separate the variables to isolate $y$ on one side and $x$ on the other side:

$\sqrt{1 - y^2} \, dy = x^2 \, dx$

Step 2: Integrate both sides

Now, we integrate both sides of the equation.

• The left-hand side involves $\sqrt{1 - y^2}$, and to integrate this, we will use the substitution method.
• The right-hand side is simpler.

Left-hand side:

To integrate $\int \sqrt{1 - y^2} \, dy$, we recognize it as the standard integral form for the arcsine function. The formula is:

$\int \sqrt{1 - y^2} \, dy = \frac{y}{2} \sqrt{1 - y^2} + \frac{1}{2} \sin^{-1}(y) + C_1$

Right-hand side:

The integral of $x^2$ with respect to $x$ is straightforward:

$\int x^2 \, dx = \frac{x^3}{3} + C_2$

Step 3: Combine the results

We now have the equation:

$\frac{y}{2} \sqrt{1 - y^2} + \frac{1}{2} \sin^{-1}(y) = \frac{x^3}{3} + C$

where $C = C_2 - C_1$ is the constant of integration.

Step 4: Apply the initial condition

We are given that $y(0) = 0$. Substituting $x = 0$ and $y = 0$ into the equation:

$\frac{0}{2} \sqrt{1 - 0^2} + \frac{1}{2} \sin^{-1}(0) = \frac{0^3}{3} + C$

This simplifies to:

$C = 0$

So the equation becomes:

$\frac{y}{2} \sqrt{1 - y^2} + \frac{1}{2} \sin^{-1}(y) = \frac{x^3}{3}$

Step 5: Solve for $y$

Unfortunately, solving this equation explicitly for $y$ in terms of $x$ is difficult because of the complexity of the terms involving $y$. However, the implicit solution we derived is the best form:

$\frac{y}{2} \sqrt{1 - y^2} + \frac{1}{2} \sin^{-1}(y) = \frac{x^3}{3}$

This is the exact formula for $y$ in terms of $x$, though it cannot be easily solved for $y$ in a simple closed form without numerical methods.

Final Answer:

The exact implicit solution to the initial value problem is:

$\frac{y}{2} \sqrt{1 - y^2} + \frac{1}{2} \sin^{-1}(y) = \frac{x^3}{3}$

Let me know if you would like further clarification or if you'd like help solving this numerically for specific values of $x$.

Here are some follow-up questions for you:

1. How would you approach solving this equation for $y$ numerically?
2. What methods could be used to solve this type of differential equation in general?
3. How do the behavior of solutions change if the initial condition was different?
4. Can we use series expansions to approximate $y(x)$ for small values of $x$?
5. What are the potential applications of this kind of differential equation?

Tip: To handle complicated implicit solutions like this one, numerical methods such as the Euler method or Runge-Kutta can be useful for approximating the solution at specific points.