The equations in the image seem to involve solutions to differential equations. Eliminating arbitrary constants from these solutions typically involves differentiating the equation multiple times, and then using conditions or relations that simplify or remove the constants.

Let's walk through the general idea of eliminating arbitrary constants for each equation:

1. Equation 5: $y = e^x(c_1 + c_2x + c_3x^2)$ This equation has three arbitrary constants $c_1, c_2, c_3$. To eliminate these constants, you would differentiate the equation multiple times (corresponding to the number of constants), and use those derivatives to form a relation that no longer involves the arbitrary constants.

2. Equation 6: $y = c_1 \sin 5x + c_2 \cos 5x$ This equation has two arbitrary constants $c_1$ and $c_2$. Similar to the previous method, you would differentiate it twice (since there are two constants) to eliminate $c_1$ and $c_2$.

3. Equation 7: $y = e^{-x}(c_1 \cos 3x + c_2 \sin 3x)$ This equation involves the arbitrary constants $c_1$ and $c_2$, so differentiating it twice would help eliminate those constants.

4. Equation 8: $y = (c_1 + c_2 x)e^x + c_3 e^{2x}$ There are three arbitrary constants $c_1, c_2, c_3$ here. Differentiating this equation three times would eliminate these constants.

Would you like to proceed with the detailed process of eliminating the arbitrary constants for one or more of these equations?