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Find the particular integral and complementary function 

Let's start solving the differential equations you uploaded. These are second-order linear non-homogeneous differential equations. Each one requires finding the **complementary function (CF)** and the **particular integral (PI)** to give the **general solution**.

### 1. Equation 14:
\[
\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = x^2
\]
This is a non-homogeneous equation. The general solution will be:
\[
y = y_c + y_p
\]
Where:
- \(y_c\) is the complementary function.
- \(y_p\) is the particular integral.

#### Complementary Function (CF):
Solve the homogeneous equation:
\[
\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 0
\]
The characteristic equation is:
\[
r^2 - 5r + 6 = 0
\]
Factor the equation:
\[
(r - 3)(r - 2) = 0
\]
So, \(r = 3\) and \(r = 2\). Thus, the complementary function is:
\[
y_c = C_1 e^{3x} + C_2 e^{2x}
\]

#### Particular Integral (PI):
For the non-homogeneous term \(x^2\), we assume the form of the particular integral as:
\[
y_p = Ax^2 + Bx + C
\]
Substitute \(y_p\) into the left-hand side of the differential equation and solve for \(A\), \(B\), and \(C\).

After substituting and solving:
- \(A = \frac{1}{6}\)
- \(B = 0\)
- \(C = 0\)

Thus, the particular integral is:
\[
y_p = \frac{x^2}{6}
\]

#### General Solution:
The general solution is:
\[
y = y_c + y_p = C_1 e^{3x} + C_2 e^{2x} + \frac{x^2}{6}
\]

---

Would you like me to proceed with the next equation (15), or provide details on any step?

Here are 5 related questions for further exploration:
1. How do you determine the form of the particular integral?
2. What happens if the non-homogeneous term is a sine or exponential function?
3. Why is the characteristic equation important in solving differential equations?
4. How does the method change for higher-order linear differential equations?
5. Can you always assume a polynomial form for the particular integral?

**Tip**: Always check if your assumed particular integral solves the non-homogeneous part by substitution, as it helps avoid errors in solving for unknown constants.

Find the particular integral and complementary function for the given differential equations.

Learn how to solve second-order non-homogeneous differential equations, including finding particular integrals and complementary functions. This page covers key techniques such as solving characteristic equations and determining the form of particular integrals based on non-homogeneous terms. Suitable for undergraduate students.

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