The differential equation
`(dy)/(dx)+x sin 2y=x^(3)cos^(2)y`
when transformed to linear form becomes

Options

A. `(dz)/(dx)+(z)/(x^(2))=x`
B. `(dz)/(dx)+zx=(x^(3))/(2)`
C. `(dz)/(dx)+2xz=x^(3)`
D. `(dz)/(dx)+(z)/(x)=x^(2)`

Explanation

To transform the given differential equation into linear form, we will follow these steps:

**Step 1: Write the original equation.**
The given differential equation is:
\[ \frac{dy}{dx} + x \sin(2y) = x^3 \cos^2(y) \]

**Step 2: Use the trigonometric identity for \(\sin(2y)\).**
We know that:
\[ \sin(2y) = 2 \sin(y) \cos(y) \]
Substituting this into the equation, we get:
\[ \frac{dy}{dx} + x (2 \sin(y) \cos(y)) = x^3 \cos^2(y) \]
This simplifies to:
\[ \frac{dy}{dx} + 2x \sin(y) \cos(y) = x^3 \cos^2(y) \]

**Step 3: Divide the entire equation by \(\cos^2(y)\).**
To isolate \(\frac{dy}{dx}\), we divide both sides by \(\cos^2(y)\):
\[ \frac{1}{\cos^2(y)} \frac{dy}{dx} + \frac{2x \sin(y)}{\cos(y)} = x^3 \]
This can be rewritten as:
\[ \sec^2(y) \frac{dy}{dx} + 2x \tan(y) = x^3 \]

**Step 4: Substitute \(z = \tan(y)\).**
Let \(z = \tan(y)\). Then, differentiating both sides with respect to \(x\):
\[ \frac{dz}{dx} = \sec^2(y) \frac{dy}{dx} \]
Substituting this into the equation gives:
\[ \frac{dz}{dx} + 2x z = x^3 \]

**Step 5: Write the final linear form.**
The equation is now in the linear form:
\[ \frac{dz}{dx} + 2xz = x^3 \]

Thus, the transformed linear form of the given differential equation is:
\[ \frac{dz}{dx} + 2xz = x^3 \]