Find the particular solution of the following :
`(dy)/(dx)=x^(3) cosec y`, given that `y(0)=0`.

Answer

To find the particular solution of the differential equation \(\frac{dy}{dx} = x^3 \csc y\) given that \(y(0) = 0\), we will follow these steps:

### Step 1: Separate the variables
We start with the equation:
\[ \frac{dy}{dx} = x^3 \csc y \]
We can rewrite \(\csc y\) as \(\frac{1}{\sin y}\). Therefore, we can separate the variables:
\[ \sin y \, dy = x^3 \, dx \]

### Step 2: Integrate both sides
Now we will integrate both sides:
\[ \int \sin y \, dy = \int x^3 \, dx \]
The left side integrates to:
\[ -\cos y \]
And the right side integrates to:
\[ \frac{x^4}{4} + C \]
Thus, we have:
\[ -\cos y = \frac{x^4}{4} + C \]

### Step 3: Solve for the constant \(C\)
To find the constant \(C\), we use the initial condition \(y(0) = 0\):
\[ -\cos(0) = \frac{0^4}{4} + C \]
Since \(\cos(0) = 1\), we have:
\[ -1 = 0 + C \implies C = -1 \]

### Step 4: Write the particular solution
Substituting \(C\) back into our equation, we get:
\[ -\cos y = \frac{x^4}{4} - 1 \]
Rearranging gives us the particular solution:
\[ -\cos y = \frac{x^4}{4} - 1 \]

### Step 5: Final form of the solution
To express the solution in a more standard form, we can write:
\[ \cos y = 1 - \frac{x^4}{4} \]

### Summary of the Particular Solution
The particular solution of the differential equation \(\frac{dy}{dx} = x^3 \csc y\) given that \(y(0) = 0\) is:
\[ \cos y = 1 - \frac{x^4}{4} \]