We are tasked with solving the differential equation:
$y' - y^2 = x^2 \quad \text{with the initial condition } y(1) = 1.$

---

Step 1: Analyze the Problem

• $y'$ is the derivative of $y$ with respect to $x$.
• Rearrange the equation into standard form:
  $y' = y^2 + x^2.$
• This is a first-order nonlinear differential equation because of the $y^2$ term.

Step 2: Solve Using Separation of Variables

The equation is not separable directly due to the form $y^2 + x^2$. Therefore, we recognize that this problem requires either a numerical approach or an advanced analytic method.

---

Step 3: Analyze for Exact or Analytical Methods

The equation $y' = y^2 + x^2$ does not match standard forms like Bernoulli's equation or exact equations. Thus, no closed-form solution exists directly. However, we can proceed by numerical methods to find $y(x)$ that satisfies the initial condition $y(1) = 1$.

---

Step 4: Implement a Numerical Solution (Euler's Method)

Using a numerical method like Euler's Method or Runge-Kutta Method would allow us to approximate the solution for $y$ over a range of $x$. If you need, I can perform numerical computations and plot the solution.

Would you like me to compute the solution numerically and plot it for you? Let me know the range of $x$ values to explore (e.g., $x = [1, 2]$).

---

Related Questions:

1. What are the steps of Euler's method to solve first-order ODEs?
2. How can Runge-Kutta methods provide a better approximation than Euler's method?
3. What is the qualitative behavior of solutions to $y' = y^2 + x^2$?
4. How can Picard's iteration method be used to solve this ODE?
5. How do nonlinear first-order ODEs differ from linear ones?

---

Tip:

If a differential equation involves terms like $y^2$, analytic solutions may not always exist. In such cases, numerical methods are invaluable for approximating solutions.