Solution of diff. eqn. `(dy)/(dx) +x sin 2y = x^(3) cos^(2)y` is ………….

Answer

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

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

### Step 2: Using the Identity for \(\sin(2y)\)
Recall that \( \sin(2y) = 2 \sin(y) \cos(y) \). Substituting this into the equation gives:
\[ \frac{dy}{dx} = x^3 \cos^2(y) - 2x \sin(y) \cos(y) \]

### Step 3: Dividing by \(\cos^2(y)\)
Next, we divide the entire equation by \( \cos^2(y) \):
\[ \frac{1}{\cos^2(y)} \frac{dy}{dx} = x^3 - 2x \tan(y) \]

### Step 4: Substituting \( \tan(y) = v \)
Let \( v = \tan(y) \). Then, we know that:
\[ \frac{dy}{dx} = \sec^2(y) \frac{dv}{dx} \]
Thus, we can rewrite the equation as:
\[ \sec^2(y) \frac{dv}{dx} = x^3 - 2x v \]

### Step 5: Expressing \(\sec^2(y)\) in Terms of \(v\)
Using the identity \( \sec^2(y) = 1 + \tan^2(y) = 1 + v^2 \), we substitute:
\[ (1 + v^2) \frac{dv}{dx} = x^3 - 2x v \]

### Step 6: Rearranging to Form a Linear Differential Equation
This can be rearranged to:
\[ \frac{dv}{dx} + \frac{2x}{1 + v^2} v = \frac{x^3}{1 + v^2} \]
This is a linear first-order differential equation in \(v\).

### Step 7: Finding the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int \frac{2x}{1 + v^2} dx} = e^{x^2} \]

### Step 8: Solving the Linear Differential Equation
We multiply through by the integrating factor:
\[ e^{x^2} \frac{dv}{dx} + 2x v e^{x^2} = x^3 e^{x^2} \]
Integrating both sides gives:
\[ \int \left( e^{x^2} \frac{dv}{dx} + 2x v e^{x^2} \right) dx = \int x^3 e^{x^2} dx \]

### Step 9: Integration
Using integration by parts on the right-hand side, we find:
\[ v e^{x^2} = \frac{1}{2} e^{x^2}(x^2 - 1) + C \]

### Step 10: Back Substituting for \(y\)
Substituting back for \(v\):
\[ \tan(y) e^{x^2} = \frac{1}{2} e^{x^2}(x^2 - 1) + C \]
Thus:
\[ \tan(y) = \frac{1}{2}(x^2 - 1) + Ce^{-x^2} \]

### Final Solution
The final solution of the differential equation is:
\[ y = \tan^{-1}\left(\frac{1}{2}(x^2 - 1) + Ce^{-x^2}\right) \]