Find the particular solution of the differential equation `(dy)/(dx)=-4x y^2`given that `y = 1`, when`x = 0`.

Explanation

To find the particular solution of the differential equation

\[ \frac{dy}{dx} = -4xy^2 \]

given that \( y = 1 \) when \( x = 0 \), we will follow these steps:

### Step 1: Separate the variables
We start by separating the variables \( y \) and \( x \).

\[ \frac{dy}{y^2} = -4x \, dx \]

### Step 2: Integrate both sides
Next, we integrate both sides. The left side integrates with respect to \( y \) and the right side with respect to \( x \).

\[ \int \frac{dy}{y^2} = \int -4x \, dx \]

The integral of \( \frac{1}{y^2} \) is \( -\frac{1}{y} \), and the integral of \( -4x \) is \( -2x^2 \). Thus, we have:

\[ -\frac{1}{y} = -2x^2 + C \]

where \( C \) is the constant of integration.

### Step 3: Rearrange the equation
Multiplying through by -1 gives:

\[ \frac{1}{y} = 2x^2 - C \]

### Step 4: Solve for \( y \)
Taking the reciprocal of both sides, we find:

\[ y = \frac{1}{2x^2 - C} \]

### Step 5: Apply the initial condition
We use the initial condition \( y = 1 \) when \( x = 0 \) to find the value of \( C \).

Substituting \( x = 0 \) into the equation:

\[ 1 = \frac{1}{2(0)^2 - C} \]

This simplifies to:

\[ 1 = \frac{1}{-C} \]

Thus, we have:

\[ -C = 1 \quad \Rightarrow \quad C = -1 \]

### Step 6: Substitute \( C \) back into the equation
Now we substitute \( C = -1 \) back into our equation for \( y \):

\[ y = \frac{1}{2x^2 - (-1)} = \frac{1}{2x^2 + 1} \]

### Conclusion
The particular solution of the differential equation is:

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

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