The equations in the image you've uploaded involve arbitrary constants, which are typically eliminated by differentiating the equation enough times to remove these constants. Below is the process for each equation shown in the image, step by step.

5. $y = e^x(c_1 + c_2 x + c_3 x^2)$

• Differentiate once: $y' = e^x(c_1 + c_2 x + c_3 x^2) + e^x(c_2 + 2c_3 x)$
• Differentiate again: $y'' = e^x(c_1 + c_2 x + c_3 x^2) + 2e^x(c_2 + 2c_3 x) + e^x(2c_3)$

After differentiating twice, the arbitrary constants $c_1$, $c_2$, and $c_3$ remain in the expression. However, further differentiations could eliminate all constants depending on the goal (e.g., finding a general solution or particular equation).

6. $y = c_1 \sin(5x) + c_2 \cos(5x)$

• Differentiate once: $y' = 5c_1 \cos(5x) - 5c_2 \sin(5x)$
• Differentiate again: $y'' = -25c_1 \sin(5x) - 25c_2 \cos(5x)$ This second-order differential equation can be written as: $y'' + 25y = 0$ Here, the arbitrary constants $c_1$ and $c_2$ have been eliminated.

7. $y = e^{-x}(c_1 \cos(3x) + c_2 \sin(3x))$

• Differentiate once using the product rule: $y' = -e^{-x}(c_1 \cos(3x) + c_2 \sin(3x)) + e^{-x}(-3c_1 \sin(3x) + 3c_2 \cos(3x))$
• Simplified: $y' = e^{-x} \left[ -c_1 \cos(3x) - c_2 \sin(3x) - 3c_1 \sin(3x) + 3c_2 \cos(3x) \right]$

You could continue differentiating further to eliminate the arbitrary constants, depending on the specific goal.

8. $y = (c_1 + c_2 x)e^x + c_3 e^{2x}$

• Differentiate once: $y' = (c_1 + c_2 x)e^x + c_2 e^x + 2c_3 e^{2x}$
• Differentiate again: $y'' = (c_1 + c_2 x)e^x + 2c_2 e^x + 4c_3 e^{2x}$

Further differentiations could be carried out if required.

Would you like more details on any specific step or further explanation on the elimination of constants?

Here are 5 related questions to consider:

1. What is the general process for solving differential equations with arbitrary constants?
2. How do we identify the order of a differential equation from a given function?
3. What conditions are necessary to fully eliminate all arbitrary constants?
4. How can we check if a given solution satisfies a differential equation?
5. What is the significance of boundary conditions in solving such equations?

Tip: To eliminate arbitrary constants, you typically need to differentiate the equation as many times as there are constants.