Solve the following differential equations.
`( 2x -10 y^2) dy + y dx =0 , y ne 0`

Answer

To solve the differential equation \( (2x - 10y^2) dy + y dx = 0 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ (2x - 10y^2) dy + y dx = 0 \]
Rearranging gives us:
\[ (2x - 10y^2) dy = -y dx \]
Dividing both sides by \(y(2x - 10y^2)\) (assuming \(y \neq 0\)):
\[ \frac{dy}{dx} = -\frac{y}{2x - 10y^2} \]

### Step 2: Finding \(\frac{dx}{dy}\)
Next, we can find \(\frac{dx}{dy}\) by taking the reciprocal:
\[ \frac{dx}{dy} = -\frac{2x - 10y^2}{y} \]
This can be rewritten as:
\[ \frac{dx}{dy} + \frac{2}{y} x = 10y \]

### Step 3: Identifying \(p\) and \(q\)
From the standard form \(\frac{dx}{dy} + p x = q\), we identify:
\[ p = \frac{2}{y}, \quad q = 10y \]

### Step 4: Finding the Integrating Factor
The integrating factor \( \mu(y) \) is given by:
\[ \mu(y) = e^{\int p \, dy} = e^{\int \frac{2}{y} \, dy} = e^{2 \ln |y|} = |y|^2 \]
Since \(y \neq 0\), we can write:
\[ \mu(y) = y^2 \]

### Step 5: Multiplying the Equation by the Integrating Factor
Multiplying the entire differential equation by \(y^2\):
\[ y^2 \frac{dx}{dy} + 2xy = 10y^3 \]

### Step 6: Recognizing the Left Side as a Derivative
The left side can be recognized as the derivative of a product:
\[ \frac{d}{dy}(xy^2) = 10y^3 \]

### Step 7: Integrating Both Sides
Integrating both sides with respect to \(y\):
\[ xy^2 = \int 10y^3 \, dy = \frac{10}{4} y^4 + C = \frac{5}{2} y^4 + C \]

### Step 8: Final Solution
Thus, we have:
\[ xy^2 = \frac{5}{2} y^4 + C \]
Rearranging gives:
\[ 2xy^2 = 5y^4 + 2C \]
Letting \(2C = C'\) (a new constant), we can express the final solution as:
\[ 2xy^2 = 5y^4 + C' \]