Let `y=(a sin x+(b+c)cosx)e^(x+d)` where a, b, c, d are parameters. The order of the differential equation formed after elimintion of parameter is ________

Answer

To solve the problem, we need to determine the order of the differential equation formed after the elimination of the parameters \( a, b, c, d \) from the given function:

\[ y = (a \sin x + (b+c) \cos x) e^{(x+d)} \]

### Step 1: Differentiate \( y \) with respect to \( x \)

To find the differential equation, we first differentiate \( y \):

\[ y' = \frac{dy}{dx} \]

Using the product rule, we differentiate \( y \):

\[ y' = \frac{d}{dx}[(a \sin x + (b+c) \cos x) e^{(x+d)}] \]

Let \( u = a \sin x + (b+c) \cos x \) and \( v = e^{(x+d)} \).

Using the product rule \( (uv)' = u'v + uv' \):

1. Differentiate \( u \):
\[ u' = a \cos x - (b+c) \sin x \]

2. Differentiate \( v \):
\[ v' = e^{(x+d)} \quad \text{(since the derivative of } e^{(x+d)} \text{ is itself)} \]

Now applying the product rule:

\[ y' = (a \cos x - (b+c) \sin x)e^{(x+d)} + (a \sin x + (b+c) \cos x)e^{(x+d)} \]

### Step 2: Simplify \( y' \)

Combining the terms:

\[ y' = e^{(x+d)} \left( a \cos x - (b+c) \sin x + a \sin x + (b+c) \cos x \right) \]

### Step 3: Differentiate \( y' \) to find \( y'' \)

Now we differentiate \( y' \) again to find \( y'' \):

\[ y'' = \frac{d}{dx}[y'] \]

Using the product rule again:

\[ y'' = \frac{d}{dx}[e^{(x+d)}(a \cos x + (b+c) \cos x - (b+c) \sin x + a \sin x)] \]

This will involve differentiating both \( e^{(x+d)} \) and the combined trigonometric terms.

### Step 4: Form the differential equation

After differentiating \( y \) and \( y' \), we can express \( y'' \) in terms of \( y \), \( y' \), and possibly other derivatives. The goal is to eliminate the parameters \( a, b, c, d \).

### Step 5: Determine the order of the differential equation

The highest derivative we calculated is \( y'' \), which indicates that the order of the differential equation is 2.

### Conclusion

Thus, the order of the differential equation formed after the elimination of the parameters is:

\[ \text{Order of the differential equation} = 2 \]

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