The image contains a differential equation that needs to be solved. Let me analyze the equation and provide a detailed solution.

Equation:
$\text{y}^{(3)} + \cos(x)\text{y}' + 2x\text{y} = \cos(2x)$

This is a third-order non-linear differential equation because of the presence of the term $\cos(x)\text{y}'$ and the non-linear nature of the right-hand side $\cos(2x)$.

Steps to Solve:

1. Identify the type of equation:

   • The equation has a third derivative $y^{(3)}$, a first derivative term $\cos(x)\text{y}'$, and the function $y$ itself multiplied by $2x$. The non-homogeneous part is $\cos(2x)$.

2. Check for linearity:

   • Despite the presence of trigonometric functions, the equation is linear in $y$, because $y^{(3)}$, $y'$, and $y$ appear with their respective coefficients, but no powers or products of these terms.

3. Solve the homogeneous equation: $y^{(3)} + \cos(x)y' + 2xy = 0$

   • To solve this, we would typically assume a solution form, possibly using a characteristic equation or a series solution, depending on the complexity of the coefficients.

4. Solve the particular solution:

   • To solve for the particular solution $y_p(x)$ to the non-homogeneous equation, one would typically use the method of undetermined coefficients or variation of parameters, assuming a form that would fit the non-homogeneous term $\cos(2x)$.

5. Combine solutions:

   • The general solution $y(x)$ would be the sum of the homogeneous solution $y_h(x)$ and the particular solution $y_p(x)$.

Detailed solution (Steps Outline):

Given that solving third-order differential equations directly might involve complex computations or assumptions, the process can include methods like:

• Assuming a polynomial or exponential form for the solution.
• Using a series expansion to handle the non-linearities introduced by $\cos(x)$.

Would you like a more detailed step-by-step breakdown of these methods, or should I attempt a specific method to solve this equation?

Relative Questions:

1. Would you like to explore how to solve third-order linear differential equations in general?
2. Are you interested in the method of undetermined coefficients?
3. Do you want to see how to solve the homogeneous part of this differential equation?
4. Would you like to understand how non-homogeneous terms influence the solution?
5. Are you interested in methods to solve non-linear differential equations?

Tip:

When dealing with higher-order differential equations, breaking down the problem into solving the homogeneous and particular solutions separately often simplifies the approach.