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:
\[ (2x - 10y^2) dy = -y dx \]
Now, we can express \( \frac{dx}{dy} \):
\[ \frac{dx}{dy} = \frac{2x - 10y^2}{-y} = \frac{10y^2 - 2x}{y} \]

### Step 2: Writing in Standard Form
We can rewrite the equation in the standard linear form \( \frac{dx}{dy} + P(y)x = Q(y) \):
\[ \frac{dx}{dy} + \frac{2}{y} x = 10y \]
Here, \( P(y) = \frac{2}{y} \) and \( Q(y) = 10y \).

### Step 3: Finding the Integrating Factor
The integrating factor \( \mu(y) \) is given by:
\[ \mu(y) = e^{\int P(y) dy} = e^{\int \frac{2}{y} dy} \]
Calculating the integral:
\[ \int \frac{2}{y} dy = 2 \ln |y| = \ln |y|^2 \]
Thus, the integrating factor becomes:
\[ \mu(y) = e^{\ln |y|^2} = |y|^2 = y^2 \quad (\text{since } y \neq 0) \]

### Step 4: Multiplying the Equation by the Integrating Factor
Now, we multiply the entire differential equation by \( y^2 \):
\[ y^2 \frac{dx}{dy} + 2xy = 10y^3 \]

### Step 5: 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 6: Integrating Both Sides
Integrating both sides with respect to \( y \):
\[ \int \frac{d}{dy}(xy^2) dy = \int 10y^3 dy \]
This gives:
\[ xy^2 = 10 \cdot \frac{y^4}{4} + C \]
\[ xy^2 = \frac{10}{4} y^4 + C \]
\[ xy^2 = \frac{5}{2} y^4 + C \]

### Step 7: Final Solution
Thus, the general solution of the differential equation is:
\[ xy^2 - \frac{5}{2} y^4 = C \]
or rearranging:
\[ xy^2 = \frac{5}{2} y^4 + C \]