`(dy)/(dx)+y/x=0`, where 'x' denotes the percentage population living in a city and 'y' denotes the area for living healthy life of population. Find the particular solution when `x=100, y=1`.

Answer

To solve the differential equation \(\frac{dy}{dx} + \frac{y}{x} = 0\), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ \frac{dy}{dx} + \frac{y}{x} = 0 \]
We can rearrange it to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Step 2: Separating Variables
Next, we separate the variables \(y\) and \(x\):
\[ \frac{dy}{y} = -\frac{dx}{x} \]

### Step 3: Integrating Both Sides
Now, we integrate both sides:
\[ \int \frac{dy}{y} = \int -\frac{dx}{x} \]
The integrals yield:
\[ \ln |y| = -\ln |x| + C \]
where \(C\) is the constant of integration.

### Step 4: Simplifying the Equation
We can simplify this equation by exponentiating both sides:
\[ |y| = e^{-\ln |x| + C} = e^{C} \cdot \frac{1}{|x|} = \frac{K}{|x|} \]
where \(K = e^{C}\) is a positive constant.

### Step 5: Removing Absolute Values
Since \(x\) represents a percentage (which is positive), we can drop the absolute values:
\[ y = \frac{K}{x} \]

### Step 6: Finding the Particular Solution
To find the particular solution, we use the initial condition \(x = 100\) and \(y = 1\):
\[ 1 = \frac{K}{100} \]
This implies:
\[ K = 100 \]

### Step 7: Writing the Final Solution
Substituting \(K\) back into the equation gives us:
\[ y = \frac{100}{x} \]

### Conclusion
Thus, the particular solution to the differential equation is:
\[ y = \frac{100}{x} \]

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