Find the differential equation by eliminating arbitrary constants from the relation `y = (c_(1) + c_(2)x)e^(x)`

Answer

To find the differential equation by eliminating the arbitrary constants \( c_1 \) and \( c_2 \) from the relation \( y = (c_1 + c_2 x)e^x \), we can follow these steps:

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

Given:
\[ y = (c_1 + c_2 x)e^x \]

We apply the product rule to differentiate:
\[ \frac{dy}{dx} = \frac{d}{dx}[(c_1 + c_2 x)e^x] = (c_1 + c_2 x)\frac{d}{dx}[e^x] + e^x\frac{d}{dx}[c_1 + c_2 x] \]
\[ = (c_1 + c_2 x)e^x + e^x(c_2) \]
\[ = (c_1 + c_2 x + c_2)e^x \]

### Step 2: Rearranging the first derivative

Now we can express this as:
\[ \frac{dy}{dx} = (c_1 + c_2 x + c_2)e^x \]

### Step 3: Substitute \( c_1 + c_2 x \) from the original equation

From the original equation, we can express \( c_1 + c_2 x \) as:
\[ c_1 + c_2 x = \frac{y}{e^x} \]

Substituting this into the derivative:
\[ \frac{dy}{dx} = \left(\frac{y}{e^x} + c_2\right)e^x \]
\[ = y + c_2 e^x \]

### Step 4: Differentiate again to eliminate \( c_2 \)

Now we differentiate \( \frac{dy}{dx} \) again:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}[y + c_2 e^x] \]
Using the product rule again:
\[ \frac{d^2y}{dx^2} = \frac{dy}{dx} + c_2 e^x \]

### Step 5: Substitute \( c_2 \) from the first derivative

From our first derivative, we can isolate \( c_2 \):
\[ c_2 = \frac{1}{e^x}\left(\frac{dy}{dx} - y\right) \]

Substituting this back into the second derivative:
\[ \frac{d^2y}{dx^2} = \frac{dy}{dx} + \left(\frac{1}{e^x}\left(\frac{dy}{dx} - y\right)\right)e^x \]
\[ = \frac{dy}{dx} + \left(\frac{dy}{dx} - y\right) \]
\[ = 2\frac{dy}{dx} - y \]

### Step 6: Rearranging to form the differential equation

Now we can rearrange this to form the differential equation:
\[ \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0 \]

### Final Differential Equation

Thus, the final differential equation is:
\[ \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0 \]

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