https://file.aiquickdraw.com/mathbot/file/1727071068246ibvcpbpb.jpg

The given equation is:

\[
\frac{dy}{dx} = \frac{y^2}{\sqrt{x+1}}
\]

This is a separable differential equation, meaning we can separate the variables \(y\) and \(x\) on opposite sides of the equation to solve it.

### Steps to solve:
1. Rewrite the equation:
\[
\frac{dy}{y^2} = \frac{dx}{\sqrt{x+1}}
\]

2. Integrate both sides:
   - The left-hand side integrates with respect to \(y\).
   - The right-hand side integrates with respect to \(x\).

Let's proceed with the integration.

#### Left side:
\[
\int \frac{1}{y^2} dy = -\frac{1}{y}
\]

#### Right side:
\[
\int \frac{1}{\sqrt{x+1}} dx = 2\sqrt{x+1}
\]

3. Combine the results:
\[
-\frac{1}{y} = 2\sqrt{x+1} + C
\]
where \(C\) is the constant of integration.

4. Finally, solve for \(y\):
\[
y = -\frac{1}{2\sqrt{x+1} + C}
\]

Thus, the general solution to the differential equation is:
\[
y = -\frac{1}{2\sqrt{x+1} + C}
\]

---

Would you like more details on any specific step, or do you have any further questions?

Here are 5 related questions for further exploration:
1. How do you solve non-separable differential equations?
2. Can this equation be solved using a different method, like an integrating factor?
3. What are the boundary conditions, if any, for a particular solution to this equation?
4. How would this solution change if there were an additional term in the equation?
5. What is the physical interpretation of such differential equations in real-world applications?

**Tip**: When solving separable differential equations, always check if you can express both sides with respect to their respective variables for easier integration.

Tentukan solusi umum dari dy/dx = y^2 / sqrt(x + 1).

Learn how to solve the separable differential equation dy/dx = y^2 / sqrt(x + 1). The solution involves separating the variables, integrating both sides, and solving for the general solution. Suitable for students in undergraduate mathematics courses.

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