The differential equation of `y= e^(2x) (A cos mx +B sin mx)` is

Options

A. `(d^(2)y)/(dx^(2))-y (dy)/(dx) + (4 + m^(2))y= 0`
B. `(d^(2)y)/(dx^(2)) +y(dy)/(dx) + (4+m^(2))y=0`
C. `(d^(2)y)/(dx^(2)) +y(dy)/(dx)-(4+m^(2))y=0`
D. None of these

Answer

To find the differential equation for the function \( y = e^{2x}(A \cos(mx) + B \sin(mx)) \), we will differentiate it twice and eliminate the constants \( A \) and \( B \).

### Step 1: Differentiate \( y \) to find \( \frac{dy}{dx} \)

Given:
\[ y = e^{2x}(A \cos(mx) + B \sin(mx)) \]

Using the product rule:
\[ \frac{dy}{dx} = \frac{d}{dx}(e^{2x}) \cdot (A \cos(mx) + B \sin(mx)) + e^{2x} \cdot \frac{d}{dx}(A \cos(mx) + B \sin(mx)) \]

Calculating \( \frac{d}{dx}(e^{2x}) \):
\[ \frac{d}{dx}(e^{2x}) = 2e^{2x} \]

Calculating \( \frac{d}{dx}(A \cos(mx) + B \sin(mx)) \):
\[ \frac{d}{dx}(A \cos(mx) + B \sin(mx)) = -Am \sin(mx) + Bm \cos(mx) \]

Putting it all together:
\[ \frac{dy}{dx} = 2e^{2x}(A \cos(mx) + B \sin(mx)) + e^{2x}(-Am \sin(mx) + Bm \cos(mx)) \]

Factoring out \( e^{2x} \):
\[ \frac{dy}{dx} = e^{2x}\left(2(A \cos(mx) + B \sin(mx)) + (-Am \sin(mx) + Bm \cos(mx))\right) \]

### Step 2: Simplify \( \frac{dy}{dx} \)

Combining the terms:
\[ \frac{dy}{dx} = e^{2x}\left((2B + Bm) \cos(mx) + (2A - Am) \sin(mx)\right) \]

### Step 3: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)

Now we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(e^{2x}\left((2B + Bm) \cos(mx) + (2A - Am) \sin(mx)\right)\right) \]

Using the product rule again:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(e^{2x})\left((2B + Bm) \cos(mx) + (2A - Am) \sin(mx)\right) + e^{2x}\frac{d}{dx}\left((2B + Bm) \cos(mx) + (2A - Am) \sin(mx)\right) \]

Calculating \( \frac{d}{dx}(e^{2x}) \):
\[ \frac{d}{dx}(e^{2x}) = 2e^{2x} \]

The derivative of the trigonometric part:
\[ \frac{d}{dx}\left((2B + Bm) \cos(mx) + (2A - Am) \sin(mx)\right) = -m(2B + Bm) \sin(mx) + m(2A - Am) \cos(mx) \]

Putting it all together:
\[ \frac{d^2y}{dx^2} = 2e^{2x}\left((2B + Bm) \cos(mx) + (2A - Am) \sin(mx)\right) + e^{2x}\left(-m(2B + Bm) \sin(mx) + m(2A - Am) \cos(mx)\right) \]

### Step 4: Combine all terms

Factoring out \( e^{2x} \):
\[ \frac{d^2y}{dx^2} = e^{2x}\left(2(2B + Bm) \cos(mx) + 2(2A - Am) \sin(mx) - m(2B + Bm) \sin(mx) + m(2A - Am) \cos(mx)\right) \]

### Step 5: Form the differential equation

Now we have:
1. \( y = e^{2x}(A \cos(mx) + B \sin(mx)) \)
2. \( \frac{dy}{dx} = e^{2x}\left((2B + Bm) \cos(mx) + (2A - Am) \sin(mx)\right) \)
3. \( \frac{d^2y}{dx^2} = e^{2x}\left(\text{combined terms}\right) \)

We can eliminate \( A \) and \( B \) to form the differential equation:
\[ \frac{d^2y}{dx^2} - 4\frac{dy}{dx} + (m^2 + 4)y = 0 \]

### Final Differential Equation
\[ \frac{d^2y}{dx^2} - 4\frac{dy}{dx} + (m^2 + 4)y = 0 \]

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