https://r2.erweima.ai/mathbot/image/1727063784116fe37e9z2.jpg

The equation shown in the image is:

\[
y \, dy = 4x \left( \sqrt{y^2 + 1} \right) dx
\]
with the initial condition \( y(0) = 1 \).

We are tasked with solving this differential equation.

### Step-by-Step Solution:

1. **Separate the variables**: 
   \[
   \frac{y \, dy}{\sqrt{y^2 + 1}} = 4x \, dx
   \]

2. **Integrate both sides**:

   For the left-hand side, use a substitution to simplify:
   Let \( u = y^2 + 1 \), so \( du = 2y \, dy \).

   Therefore, the integral on the left becomes:
   \[
   \int \frac{y \, dy}{\sqrt{y^2 + 1}} = \frac{1}{2} \int \frac{du}{\sqrt{u}} = \sqrt{u} = \sqrt{y^2 + 1}
   \]

   The integral on the right is:
   \[
   \int 4x \, dx = 2x^2
   \]

3. **Combine the results**:
   \[
   \sqrt{y^2 + 1} = 2x^2 + C
   \]

4. **Apply the initial condition** \( y(0) = 1 \):

   Plugging \( x = 0 \) and \( y = 1 \) into the equation:
   \[
   \sqrt{1^2 + 1} = 2(0)^2 + C \quad \Rightarrow \quad \sqrt{2} = C
   \]

   So, \( C = \sqrt{2} \).

5. **Final solution**:
   \[
   \sqrt{y^2 + 1} = 2x^2 + \sqrt{2}
   \]
   
   Squaring both sides:
   \[
   y^2 + 1 = (2x^2 + \sqrt{2})^2
   \]
   Expanding the square:
   \[
   y^2 + 1 = 4x^4 + 4x^2\sqrt{2} + 2
   \]

   Therefore, the solution to the differential equation is:
   \[
   y^2 = 4x^4 + 4x^2\sqrt{2} + 1
   \]
   Taking the square root:
   \[
   y = \sqrt{4x^4 + 4x^2\sqrt{2} + 1}
   \]

---

Would you like further details on any of the steps? Here are some related questions:

1. How is the substitution method applied in integration?
2. What other methods could be used to solve similar differential equations?
3. Can this equation be solved using numerical methods instead of analytical ones?
4. How do initial conditions affect the solution of a differential equation?
5. How does squaring affect the range and behavior of the solution?

**Tip**: When integrating using substitution, always remember to adjust both the function and the differential accordingly to simplify the process!

y dy = 4x (y^2 + 1)^(1/2) dx with the condition y(0) = 1.

Learn how to solve the differential equation y dy = 4x (y^2 + 1)^(1/2) dx with an initial condition y(0) = 1. This problem involves separation of variables, integration using substitution, and applying initial conditions. A step-by-step solution is provided, suitable for students at an undergraduate level.

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