The solution of the differential equaiton `(ydx+xdy)/(ydx-xdy)=(x^(2)e^(xy))/(y^(4))` satisfying y(0) = 1, is

Options

A. `x^(3)=3y^(3)(-1+e^(-xy))`
B. `x^(3)=3y^(3)(1-e^(-xy))`
C. `x^(3)=3y^(3)(-1+e^(xy))`
D. `x^(3)=3y^(3)(1-e^(xy))`

Explanation

To solve the given differential equation:

\[ \frac{ydx + xdy}{ydx - xdy} = \frac{x^2 e^{xy}}{y^4} \]

with the initial condition \( y(0) = 1 \), we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fraction:

\[ (ydx + xdy) \cdot y^4 = (x^2 e^{xy}) \cdot (ydx - xdy) \]

### Step 2: Rearranging the Equation
Expanding both sides gives:

\[ y^4(ydx + xdy) = x^2 e^{xy} (ydx - xdy) \]

This can be rewritten as:

\[ y^5 dx + xy^4 dy = x^2 e^{xy} y dx - x^3 e^{xy} dy \]

### Step 3: Grouping Terms
Rearranging the terms, we have:

\[ y^5 dx + x^3 e^{xy} dy + xy^4 dy = x^2 e^{xy} y dx \]

### Step 4: Substituting Variables
Now, we can simplify the equation by substituting \( xy = t \). Differentiating both sides gives:

\[ x dy + y dx = dt \]

Thus, we can express \( y dx + x dy \) as \( dt \).

### Step 5: Rewrite the Equation
Now substituting back into the equation, we have:

\[ y^4 dt = x^2 e^t (ydx - xdy) \]

### Step 6: Further Simplification
We can rewrite \( ydx - xdy \) using the quotient rule:

\[ \frac{dy}{dx} = \frac{y^2}{x^2} \Rightarrow dy = \frac{y^2}{x^2} dx \]

### Step 7: Integrating Both Sides
Now we can integrate both sides:

\[ \int \frac{dt}{e^t} = \int p^2 dp \]

Where \( p = \frac{x}{y} \).

### Step 8: Solve the Integrals
The left side integrates to:

\[ -e^{-t} = \frac{p^3}{3} + C \]

### Step 9: Substitute Back
Substituting back \( t = xy \) and \( p = \frac{x}{y} \):

\[ -e^{-xy} = \frac{x^3}{3y^3} + C \]

### Step 10: Apply Initial Condition
Using the initial condition \( y(0) = 1 \):

\[ -e^{0} = \frac{0^3}{3 \cdot 1^3} + C \Rightarrow -1 = C \]

### Step 11: Final Equation
Substituting \( C \) back into the equation gives:

\[ -e^{-xy} = \frac{x^3}{3y^3} - 1 \]

Multiplying through by \(-3y^3\):

\[ 3y^3 e^{-xy} = x^3 - 3y^3 \]

### Step 12: Rearranging the Final Form
Thus, we can express the final solution as:

\[ x^3 = 3y^3(1 - e^{-xy}) \]

This is the required solution of the differential equation.

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