Solution of differential equation `(dy ) /( dx) +(x ) /( 1 - x^2) y= x sqrt(y) ` is

Options

A. a) `3sqrty+(1-x^(2))=c(1-x^(2))^(1//4)`
B. b) `(3)/(2)sqrty+(1-x^(2))=c(1-x^(2))^(3//2)`
C. c) `3sqrty-(1-x^(2))=c(1-x^(2))^(1//4)`
D. d) None of these

Answer

To solve the differential equation

\[ \frac{dy}{dx} + \frac{x}{1 - x^2} y = x \sqrt{y}, \]

we can follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:

\[ \frac{dy}{dx} + \frac{x}{1 - x^2} y = x \sqrt{y}. \]

### Step 2: Divide by \(\sqrt{y}\)
To simplify the equation, we divide through by \(\sqrt{y}\):

\[ \frac{1}{\sqrt{y}} \frac{dy}{dx} + \frac{x}{1 - x^2} \sqrt{y} = x. \]

### Step 3: Substitute \(t = \sqrt{y}\)
Let \(t = \sqrt{y}\). Then, we have:

\[ y = t^2 \quad \text{and} \quad \frac{dy}{dx} = 2t \frac{dt}{dx}. \]

Substituting this into the equation gives:

\[ \frac{1}{t} (2t \frac{dt}{dx}) + \frac{x}{1 - x^2} t = x. \]

This simplifies to:

\[ 2 \frac{dt}{dx} + \frac{x}{1 - x^2} t = x. \]

### Step 4: Rearrange the Equation
Now, divide the entire equation by 2:

\[ \frac{dt}{dx} + \frac{x}{2(1 - x^2)} t = \frac{x}{2}. \]

### Step 5: Identify \(p\) and \(q\)
This is a linear first-order differential equation in \(t\) where:

- \(p = \frac{x}{2(1 - x^2)}\)
- \(q = \frac{x}{2}\)

### Step 6: Find the Integrating Factor
The integrating factor \(I\) is given by:

\[ I = e^{\int p \, dx} = e^{\int \frac{x}{2(1 - x^2)} \, dx}. \]

To compute the integral, we use the substitution \(z = 1 - x^2\), leading to \(dz = -2x \, dx\). Thus, we have:

\[ \int \frac{x}{2(1 - x^2)} \, dx = -\frac{1}{4} \ln |1 - x^2| + C. \]

Therefore, the integrating factor becomes:

\[ I = e^{-\frac{1}{4} \ln |1 - x^2|} = (1 - x^2)^{-\frac{1}{4}}. \]

### Step 7: Multiply the Equation by the Integrating Factor
Multiply the entire differential equation by the integrating factor:

\[ (1 - x^2)^{-\frac{1}{4}} \left( \frac{dt}{dx} + \frac{x}{2(1 - x^2)} t \right) = (1 - x^2)^{-\frac{1}{4}} \cdot \frac{x}{2}. \]

### Step 8: Solve the Equation
The left-hand side can be expressed as:

\[ \frac{d}{dx} \left( t (1 - x^2)^{-\frac{1}{4}} \right) = \frac{x}{2(1 - x^2)^{\frac{1}{4}}}. \]

Integrating both sides gives:

\[ t (1 - x^2)^{-\frac{1}{4}} = \int \frac{x}{2(1 - x^2)^{\frac{1}{4}}} \, dx + C. \]

### Step 9: Substitute Back for \(y\)
After integrating and simplifying, we substitute back \(t = \sqrt{y}\):

\[ \sqrt{y} (1 - x^2)^{-\frac{1}{4}} = \text{(integrated result)} + C. \]

### Final Step: Rearranging for \(y\)
Finally, we rearrange to find \(y\):

\[ 3 \sqrt{y} + 1 - x^2 = C (1 - x^2)^{\frac{1}{4}}. \]

This gives us the solution to the differential equation.