Consider the differential equation
𝑑
𝑦
𝑑
𝑥
=
𝑦
2
sin
⁡
(
𝑥
)
dx
dy
​
=y
2
sin(x)
with the initial condition
𝑦
(
0
)
=
1
y(0)=1

To solve the differential equation

\[
\frac{dy}{dx} = y^2 \sin(x)
\]

with the initial condition \( y(0) = 1 \), we can use the method of separation of variables. Here are the steps:

1. **Separate the variables**:
   \[
   \frac{dy}{y^2} = \sin(x) \, dx
   \]

2. **Integrate both sides**:
   \[
   \int \frac{dy}{y^2} = \int \sin(x) \, dx
   \]

   The integral on the left side is:
   \[
   \int y^{-2} \, dy = -y^{-1} = -\frac{1}{y}
   \]

   The integral on the right side is:
   \[
   \int \sin(x) \, dx = -\cos(x)
   \]

   So, we have:
   \[
   -\frac{1}{y} = -\cos(x) + C
   \]

3. **Solve for \( y \)**:
   \[
   \frac{1}{y} = \cos(x) - C
   \]
   \[
   y = \frac{1}{\cos(x) - C}
   \]

4. **Apply the initial condition \( y(0) = 1 \)**:
   \[
   1 = \frac{1}{\cos(0) - C}
   \]
   \[
   1 = \frac{1}{1 - C}
   \]
   \[
   1 - C = 1
   \]
   \[
   C = 0
   \]

   Therefore, the solution to the differential equation with the given initial condition is:
   \[
   y = \frac{1}{\cos(x)}
   \]

Would you like more details or have any questions?

Here are 5 numbered questions for you to ask next:
1. Can you explain the method of separation of variables in more detail?
2. How do you solve initial value problems for differential equations?
3. What are some common methods for solving first-order differential equations?
4. Can you show another example of solving a differential equation using separation of variables?
5. How do you verify the solution of a differential equation?

**Tip:** When solving differential equations, always remember to apply initial conditions to find the specific solution that satisfies the given problem.

Learn how to solve the differential equation dy/dx = y^2 * sin(x) with the initial condition y(0) = 1 using separation of variables. Detailed step-by-step solution included.

Math Problem Statement

Consider the differential equation 𝑑 𝑦 𝑑 𝑥

𝑦 2 sin ⁡ ( 𝑥 ) dx dy =y 2 sin(x) with the initial condition 𝑦 ( 0 )

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