The solution of the differential equation `(3sin^(2)xcosx) y^(2)dx+2ysin^(3)xdy=sinx dx` (where, C is an arbitrary constant)

Options

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

Answer

To solve the differential equation

\[ (3\sin^2 x \cos x) y^2 \, dx + (2y \sin^3 x) \, dy = \sin x \, dx, \]

we will follow these steps:

### Step 1: Rearranging the Equation
First, we can rearrange the equation to isolate the terms involving \(dx\) and \(dy\):

\[ (3\sin^2 x \cos x) y^2 \, dx + (2y \sin^3 x) \, dy - \sin x \, dx = 0. \]

This can be rewritten as:

\[ (3\sin^2 x \cos x y^2 - \sin x) \, dx + (2y \sin^3 x) \, dy = 0. \]

### Step 2: Identifying the Form
We can identify this as a total differential equation. We can express it in the form:

\[ M(x, y) \, dx + N(x, y) \, dy = 0, \]

where

\[ M(x, y) = 3\sin^2 x \cos x y^2 - \sin x, \]
\[ N(x, y) = 2y \sin^3 x. \]

### Step 3: Finding the Partial Derivatives
Next, we need to check if the equation is exact by finding the partial derivatives:

\[ \frac{\partial M}{\partial y} = 6\sin^2 x \cos x y, \]
\[ \frac{\partial N}{\partial x} = 2y \cdot 3\sin^2 x \cos x = 6y \sin^2 x \cos x. \]

Since

\[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}, \]

the equation is exact.

### Step 4: Finding the Potential Function
To find the potential function \(F(x, y)\), we integrate \(M\) with respect to \(x\):

\[ F(x, y) = \int (3\sin^2 x \cos x y^2 - \sin x) \, dx. \]

Calculating the integral:

\[ F(x, y) = y^2 \cdot \sin^3 x - \cos x + g(y), \]

where \(g(y)\) is an arbitrary function of \(y\).

Next, we differentiate \(F\) with respect to \(y\):

\[ \frac{\partial F}{\partial y} = 2y \sin^3 x + g'(y). \]

Setting this equal to \(N\):

\[ 2y \sin^3 x + g'(y) = 2y \sin^3 x. \]

This implies \(g'(y) = 0\), so \(g(y) = C\), where \(C\) is a constant.

### Step 5: Final Solution
Thus, the potential function is:

\[ F(x, y) = y^2 \sin^3 x - \cos x = C. \]

Rearranging gives us the solution to the differential equation:

\[ y^2 \sin^3 x - \cos x = C. \]