Find the differential equation whose general solution is given by `y=(c_(1)+c_(2))cos(x+c_(3))-c_(4)e^(x+c_(5))`, where `c_(1),c_(2), c_(3), c_(4), c_(5)` are arbitary constants.

Explanation

To find the differential equation whose general solution is given by

\[ y = (c_1 + c_2) \cos(x + c_3) - c_4 e^{(x + c_5)}, \]

where \(c_1, c_2, c_3, c_4, c_5\) are arbitrary constants, we will follow these steps:

### Step 1: Rewrite the Equation
We can simplify the expression by letting:
- \(a = c_1 + c_2\) (a new constant)
- \(b = c_3\) (another constant)
- \(c = c_4 e^{c_5}\) (a new constant)

Thus, we can rewrite the equation as:

\[ y = a \cos(x) + b - c e^x. \]

### Step 2: Differentiate the Equation
Now we differentiate \(y\) with respect to \(x\):

\[ \frac{dy}{dx} = -a \sin(x) + 0 - c e^x = -a \sin(x) - c e^x. \]

### Step 3: Differentiate Again
Next, we differentiate again to find the second derivative:

\[ \frac{d^2y}{dx^2} = -a \cos(x) - c e^x. \]

### Step 4: Differentiate a Third Time
Now we differentiate a third time to find the third derivative:

\[ \frac{d^3y}{dx^3} = a \sin(x) - c e^x. \]

### Step 5: Expressing in Terms of \(y\)
Now we will express the second and third derivatives in terms of \(y\).

From the original equation, we have:

\[ y = a \cos(x) + b - c e^x. \]

Rearranging gives:

\[ c e^x = a \cos(x) + b - y. \]

Substituting this into the second derivative:

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

This simplifies to:

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

### Step 6: Substitute Back
Now we can express \(y\) in terms of \(d^2y/dx^2\):

\[ d^2y/dx^2 + 2a \cos(x) + b - y = 0. \]

### Step 7: Substitute for \(d^3y/dx^3\)
Now we can express the third derivative:

\[ \frac{d^3y}{dx^3} + c e^x = a \sin(x). \]

Substituting \(c e^x\) gives:

\[ \frac{d^3y}{dx^3} + (a \cos(x) + b - y) = a \sin(x). \]

### Step 8: Form the Final Differential Equation
Combining these results leads to the final differential equation:

\[ \frac{d^3y}{dx^3} - \frac{d^2y}{dx^2} + \frac{dy}{dx} - y = 0. \]

This is the required differential equation.