Solution of differential equation `x^(2)y - x^(3) (dy)/(dx)=y^(4) cos x` is

Options

A. `x^(2)y^(-3) = 2 sin x + c`
B. `x^(2)y^(-3) = 3 cos x + c`
C. `x^(3)y^(-3) = 3 sin x + c`
D. `x^(2)y^(3) = 3 sin x + c x^(2) y `

Explanation

To solve the differential equation \( x^2 y - x^3 \frac{dy}{dx} = y^4 \cos x \), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the given equation to isolate the derivative term:
\[ x^3 \frac{dy}{dx} = x^2 y - y^4 \cos x \]

### Step 2: Dividing by \(x^3 y^4\)
Next, we divide the entire equation by \( -x^3 y^4 \):
\[ -\frac{dy}{dx} + \frac{y}{x} = -\frac{\cos x}{y^3} \]

### Step 3: Introducing a Substitution
Let \( v = \frac{1}{y^3} \). Then, we have:
\[ \frac{dy}{dx} = -\frac{3v^2}{dx} \]
Substituting this into our equation gives:
\[ -\left(-\frac{3v^2}{dx}\right) + \frac{1}{x} \cdot \frac{1}{v^{\frac{1}{3}}} = -\cos x \]

### Step 4: Rearranging the Equation Again
Rearranging this gives us:
\[ \frac{3v^2}{dx} + \frac{3v}{x} = \cos x \]

### Step 5: Finding the Integrating Factor
The equation can be written in the standard form:
\[ \frac{dv}{dx} + \frac{3}{x} v = \frac{\cos x}{3} \]
To solve this, we need to find the integrating factor \( \mu(x) \):
\[ \mu(x) = e^{\int \frac{3}{x} dx} = e^{3 \ln x} = x^3 \]

### Step 6: Multiplying by the Integrating Factor
Now, we multiply the entire equation by the integrating factor:
\[ x^3 \frac{dv}{dx} + 3x^2 v = x^3 \cdot \frac{\cos x}{3} \]

### Step 7: Integrating Both Sides
The left-hand side can be expressed as the derivative of a product:
\[ \frac{d}{dx}(x^3 v) = \frac{x^3 \cos x}{3} \]
Integrating both sides gives:
\[ x^3 v = \int \frac{x^3 \cos x}{3} dx + C \]

### Step 8: Solving the Integral
Using integration by parts or a table of integrals, we can find:
\[ \int x^3 \cos x \, dx = \text{(result from integration)} \]
Let’s denote this result as \( I(x) \):
\[ x^3 v = \frac{I(x)}{3} + C \]

### Step 9: Substituting Back for \(y\)
Recall that \( v = \frac{1}{y^3} \), so:
\[ x^3 \cdot \frac{1}{y^3} = \frac{I(x)}{3} + C \]
Rearranging gives:
\[ y^3 = \frac{3x^3}{I(x) + 3C} \]

### Final Solution
Thus, the solution of the differential equation is:
\[ y^3 = \frac{3x^3}{\int x^3 \cos x \, dx + C} \]