To find the **particular integral** for the differential equation \((d^2 + 5d + 6)y = e^x\), we will follow the method for solving non-homogeneous linear differential equations.

### Given equation:
\[
(d^2 + 5d + 6)y = e^x
\]
Here, \( d \) denotes the differential operator \(\frac{d}{dx}\), and \( e^x \) is the non-homogeneous term.

#### Step 1: Characteristic Equation for the Homogeneous Part
First, solve the homogeneous part:
\[
d^2 + 5d + 6 = 0
\]
Factor this quadratic equation:
\[
(d + 2)(d + 3) = 0
\]
So, the roots of the characteristic equation are \( d = -2 \) and \( d = -3 \).

The **complementary function** (general solution to the homogeneous equation) is:
\[
y_c = C_1 e^{-2x} + C_2 e^{-3x}
\]

#### Step 2: Particular Integral (PI)
Now, we find the particular integral. The non-homogeneous term is \( e^x \). Since \( e^x \) is not a solution to the homogeneous equation (the roots were -2 and -3), we try a particular solution of the form:
\[
y_p = A e^x
\]
where \( A \) is a constant to be determined.

#### Step 3: Apply the Operator
We apply the operator \( d^2 + 5d + 6 \) to the trial solution \( y_p = A e^x \):
\[
d(y_p) = A e^x \quad \text{and} \quad d^2(y_p) = A e^x
\]
So, applying the operator to \( y_p \):
\[
(d^2 + 5d + 6)y_p = A e^x + 5A e^x + 6A e^x = (1 + 5 + 6)A e^x = 12A e^x
\]

#### Step 4: Solve for \( A \)
Set this equal to the non-homogeneous term \( e^x \):
\[
12A e^x = e^x
\]
So, \( A = \frac{1}{12} \).

Thus, the particular integral is:
\[
y_p = \frac{1}{12} e^x
\]

### Step 5: General Solution
The general solution to the differential equation is the sum of the complementary function and the particular integral:
\[
y = y_c + y_p = C_1 e^{-2x} + C_2 e^{-3x} + \frac{1}{12} e^x
\]

This is the required solution.

---

Would you like further details or have any questions?

### Related Questions:
1. How do we handle non-homogeneous terms that involve trigonometric functions like \(\sin x\) or \(\cos x\)?
2. What if \( e^x \) was a solution to the homogeneous equation? How would the method change?
3. How do you solve second-order differential equations with repeated roots in the characteristic equation?
4. How do we find a particular integral when the non-homogeneous term is a polynomial function?
5. Can we apply variation of parameters to solve this problem instead of the method of undetermined coefficients?

### Tip:
Always check if the non-homogeneous term is a solution of the homogeneous equation before assuming the form of the particular integral.

Solve the differential equation (d²+5d+6)y = e^x using the method of undetermined coefficients. This solution involves finding the characteristic equation, the complementary function, and the particular integral. Suitable for advanced high school and undergraduate students.

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