General solution of `(dy)/(dx)+(3y^(2))/(2x)=0` is

Options

A. `3logx+(2)/(y)=c`
B. `2logx+(3)/(y)=c`
C. `3logx-(2)/(4)=c`
D. None of these

Answer

To solve the differential equation \(\frac{dy}{dx} + \frac{3y^2}{2x} = 0\), we will use the method of separation of variables. Here are the steps to find the general solution:

### Step 1: Rearrange the equation
We start with the given differential equation:
\[ \frac{dy}{dx} + \frac{3y^2}{2x} = 0 \]
We can rearrange this to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{3y^2}{2x} \]

### Step 2: Separate the variables
Next, we separate the variables \(y\) and \(x\):
\[ \frac{dy}{y^2} = -\frac{3}{2} \frac{dx}{x} \]

### Step 3: Integrate both sides
Now, we integrate both sides. The left side integrates as follows:
\[ \int \frac{dy}{y^2} = -\frac{1}{y} \]
And the right side integrates as:
\[ \int -\frac{3}{2} \frac{dx}{x} = -\frac{3}{2} \ln |x| + C \]
Combining these results, we have:
\[ -\frac{1}{y} = -\frac{3}{2} \ln |x| + C \]

### Step 4: Simplify the equation
Multiplying through by -1 gives:
\[ \frac{1}{y} = \frac{3}{2} \ln |x| - C \]
We can rewrite \(C\) as \(-C\) (let's call it \(C'\)):
\[ \frac{1}{y} = \frac{3}{2} \ln |x| + C' \]

### Step 5: Solve for \(y\)
Taking the reciprocal gives us:
\[ y = \frac{1}{\frac{3}{2} \ln |x| + C'} \]

### Step 6: General solution
Thus, the general solution of the differential equation is:
\[ y = \frac{1}{\frac{3}{2} \ln |x| + C} \]
where \(C\) is an arbitrary constant.