The image shows a mathematical problem involving a differential equation. Let's extract the problem from the image:

The equation given is: $y' = \frac{x}{2y}, \quad y(0) = 1$

This looks like a first-order ordinary differential equation (ODE) with an initial condition. I'll solve it step by step.

Step 1: Separate variables

We need to separate the variables $y$ and $x$. Start by multiplying both sides by $2y$:

$2y y' = x$

Now, express this as:

$2y \frac{dy}{dx} = x$

Step 2: Integration

Separate the variables:

$2y \, dy = x \, dx$

Integrate both sides:

$\int 2y \, dy = \int x \, dx$

This gives:

$y^2 = \frac{x^2}{2} + C$

Step 3: Apply the initial condition

The initial condition is $y(0) = 1$. Substitute $x = 0$ and $y = 1$ into the equation:

$1^2 = \frac{0^2}{2} + C \quad \Rightarrow \quad C = 1$

Step 4: Final solution

Thus, the solution to the differential equation is:

$y^2 = \frac{x^2}{2} + 1$

Or equivalently:

$y = \sqrt{\frac{x^2}{2} + 1}$

Would you like more details on any of the steps? Here are 5 related questions for practice:

1. What type of differential equation was solved here?
2. How can we verify the solution of this differential equation?
3. What are the general methods for solving first-order separable ODEs?
4. Can this solution be used to model physical phenomena? If so, in what context?
5. What would the solution look like if the initial condition was $y(0) = 2$?

Tip: When solving ODEs, always check if the equation is separable, linear, or exact—it helps determine the best method for solving it!