Find the particular solution of the following :
`(dy)/(dx)= y tan x`, given that`y=1` when `x=0`.

Answer

To find the particular solution of the differential equation \(\frac{dy}{dx} = y \tan x\) given that \(y = 1\) when \(x = 0\), we will follow these steps:

### Step 1: Separate the Variables
We start by rewriting the equation to separate the variables \(y\) and \(x\):
\[ \frac{dy}{y} = \tan x \, dx \]

### Step 2: Integrate Both Sides
Next, we integrate both sides:
\[ \int \frac{dy}{y} = \int \tan x \, dx \]
The left side integrates to:
\[ \log |y| + C_1 \]
And the right side integrates to:
\[ -\log |\cos x| + C_2 \]
Combining these, we have:
\[ \log |y| = -\log |\cos x| + C \]
where \(C = C_2 - C_1\).

### Step 3: Simplify the Equation
We can rewrite the equation using properties of logarithms:
\[ \log |y| + \log |\cos x| = C \]
This simplifies to:
\[ \log |y \cos x| = C \]
Exponentiating both sides gives:
\[ |y \cos x| = e^C \]
Let \(k = e^C\), then:
\[ y \cos x = k \]

### Step 4: Solve for \(y\)
We can express \(y\) in terms of \(x\):
\[ y = \frac{k}{\cos x} \]

### Step 5: Use the Initial Condition
Now we use the initial condition \(y = 1\) when \(x = 0\):
\[ 1 = \frac{k}{\cos(0)} \implies 1 = \frac{k}{1} \implies k = 1 \]

### Step 6: Write the Particular Solution
Substituting \(k\) back into the equation, we find:
\[ y = \frac{1}{\cos x} \]
Thus, the particular solution is:
\[ y = \sec x \]

### Summary of the Solution
The particular solution of the differential equation \(\frac{dy}{dx} = y \tan x\) with the initial condition \(y = 1\) when \(x = 0\) is:
\[ y = \sec x \]