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.