The given image appears to show a third-order linear non-homogeneous differential equation of the form:

$\frac{d^3 y}{dx^3} + 2\frac{d^2 y}{dx^2} + \frac{dy}{dx} = e^{2x} + \sin(2x)$

To solve for the particular integral, we need to address the non-homogeneous part of the equation, which includes $e^{2x}$ and $\sin(2x)$. Here's the typical procedure to find the particular integral for such an equation:

1. Solve the Complementary Function (CF)

First, solve the homogeneous part of the equation:

$\frac{d^3 y}{dx^3} + 2\frac{d^2 y}{dx^2} + \frac{dy}{dx} = 0$

This will involve solving the characteristic equation, which is obtained by assuming $y = e^{rx}$, substituting into the homogeneous equation, and solving for $r$.

2. Particular Integral (PI) for $e^{2x}$

For the term $e^{2x}$, assume a particular solution of the form: $y_p = A e^{2x}$ where $A$ is a constant to be determined. Substitute this into the left-hand side of the differential equation and solve for $A$.

3. Particular Integral (PI) for $\sin(2x)$

For the $\sin(2x)$ term, assume a particular solution of the form: $y_p = B \sin(2x) + C \cos(2x)$ where $B$ and $C$ are constants to be determined. Substitute this into the differential equation and solve for $B$ and $C$.

Once you find the complementary function and the particular integral, the general solution will be:

$y = y_{\text{CF}} + y_{\text{PI}}$

Would you like to proceed with the detailed solution for the complementary function and particular integral, or focus on a specific part of this process?

Related Questions:

1. How do you derive the characteristic equation for a third-order differential equation?
2. What is the general method for solving non-homogeneous differential equations?
3. How do you handle terms like $e^{2x}$ when solving differential equations?
4. What approach is used to solve for a particular integral with trigonometric terms?
5. Can you explain how to combine complementary and particular solutions?

Tip:

For higher-order differential equations, the superposition principle allows you to combine solutions corresponding to different non-homogeneous terms independently.