Obtain the differential equation by eliminating 'a' and 'b' from the equation :
`y= e^(x)(a cos 2x + b sin 2x)`.

Answer

To obtain the differential equation by eliminating the constants 'a' and 'b' from the equation

\[ y = e^x (a \cos 2x + b \sin 2x), \]

we will follow these steps:

### Step 1: Differentiate the given equation once with respect to \( x \).

Using the product rule, we differentiate:

\[ \frac{dy}{dx} = e^x (a \cos 2x + b \sin 2x) + e^x \left( \frac{d}{dx}(a \cos 2x + b \sin 2x) \right). \]

Now, we differentiate \( a \cos 2x + b \sin 2x \):

\[ \frac{d}{dx}(a \cos 2x + b \sin 2x) = -2a \sin 2x + 2b \cos 2x. \]

Thus, we can write:

\[ \frac{dy}{dx} = e^x (a \cos 2x + b \sin 2x) + e^x (-2a \sin 2x + 2b \cos 2x). \]

Combining the terms gives:

\[ \frac{dy}{dx} = e^x \left( (a + 2b) \cos 2x + (b - 2a) \sin 2x \right). \]

### Step 2: Differentiate again to find the second derivative.

Now we differentiate \( \frac{dy}{dx} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( e^x \left( (a + 2b) \cos 2x + (b - 2a) \sin 2x \right) \right). \]

Applying the product rule again:

\[ \frac{d^2y}{dx^2} = e^x \left( (a + 2b) \cos 2x + (b - 2a) \sin 2x \right) + e^x \left( \frac{d}{dx} \left( (a + 2b) \cos 2x + (b - 2a) \sin 2x \right) \right). \]

Now, differentiate \( (a + 2b) \cos 2x + (b - 2a) \sin 2x \):

\[ \frac{d}{dx} \left( (a + 2b) \cos 2x + (b - 2a) \sin 2x \right) = -2(a + 2b) \sin 2x + 2(b - 2a) \cos 2x. \]

Thus, we have:

\[ \frac{d^2y}{dx^2} = e^x \left( (a + 2b) \cos 2x + (b - 2a) \sin 2x \right) + e^x \left( -2(a + 2b) \sin 2x + 2(b - 2a) \cos 2x \right). \]

Combining these gives:

\[ \frac{d^2y}{dx^2} = e^x \left( (a + 2b + 2(b - 2a)) \cos 2x + (b - 2a - 2(a + 2b)) \sin 2x \right). \]

### Step 3: Substitute \( y \) and \( \frac{dy}{dx} \) into the equation.

Now we can express the second derivative in terms of \( y \) and \( \frac{dy}{dx} \). We have:

\[ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + 5y = 0. \]

This is the required differential equation after eliminating \( a \) and \( b \).

### Final Result:
The differential equation is:

\[ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + 5y = 0. \]

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