https://file.aiquickdraw.com/mathbot/file/1725544074334xq21crah.jpg

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!

y' = x / (2y), y(0) = 1. Solve the differential equation.

Learn how to solve the separable differential equation y' = x / 2y with an initial condition y(0) = 1. The solution involves separating variables and integrating both sides. Ideal for undergraduate students studying calculus or differential equations.

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