The solution of the differential equation `x^(3)(dy)/(dx)+4x^(2) tany=e^(x) secy` satisfying `y(1)=0`, is

Options

A. `tany=(x-2)e^(x)logx`
B. `siny=e^(x)(x-1)x^(-4)`
C. `tany=(x-1)e^(x)x^(-3)`
D. `siny=e^(x)(x-1)x^(-3)`

Explanation

To solve the given differential equation

\[ x^3 \frac{dy}{dx} + 4x^2 \tan y = e^x \sec y \]

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

### Step 1: Rearranging the Equation
First, we will divide the entire equation by \( x^3 \) to simplify it:

\[ \frac{dy}{dx} + \frac{4}{x} \tan y = \frac{e^x \sec y}{x^3} \]

### Step 2: Multiplying by \( \cos y \)
Next, we multiply through by \( \cos y \):

\[ \cos y \frac{dy}{dx} + \frac{4}{x} \sin y = \frac{e^x}{x^3} \]

### Step 3: Rearranging for Integration
Now, we can rearrange this equation for integration:

\[ \cos y \frac{dy}{dx} = \frac{e^x}{x^3} - \frac{4}{x} \sin y \]

### Step 4: Multiplying by \( x^4 \)
To facilitate integration, we multiply the entire equation by \( x^4 \):

\[ x^4 \cos y \frac{dy}{dx} + 4x^3 \sin y = x e^x \]

### Step 5: Recognizing the Left Side as a Derivative
The left-hand side can be recognized as the derivative of a product:

\[ \frac{d}{dx}(x^4 \sin y) = x e^x \]

### Step 6: Integrating Both Sides
Now we integrate both sides:

\[ \int \frac{d}{dx}(x^4 \sin y) \, dx = \int x e^x \, dx \]

The integral on the left gives us:

\[ x^4 \sin y = \int x e^x \, dx \]

Using integration by parts, we find:

\[ \int x e^x \, dx = e^x (x - 1) + C \]

Thus, we have:

\[ x^4 \sin y = e^x (x - 1) + C \]

### Step 7: Solving for \( \sin y \)
Now, we can express \( \sin y \):

\[ \sin y = \frac{e^x (x - 1) + C}{x^4} \]

### Step 8: Applying the Initial Condition
We know that \( y(1) = 0 \), which implies \( \sin(0) = 0 \). Substituting \( x = 1 \):

\[ 0 = \frac{e^1 (1 - 1) + C}{1^4} \]

This simplifies to:

\[ 0 = C \]

### Step 9: Final Solution
Substituting \( C = 0 \) back into the equation gives us:

\[ \sin y = \frac{e^x (x - 1)}{x^4} \]

### Conclusion
The solution to the differential equation is:

\[ \sin y = \frac{e^x (x - 1)}{x^4} \]