Question

Use undetermined coefficients to find the particular solution to

25y'' - 5y' - 12y = 3e⁴ᵗ
Y(t) =
!!! The challenge is that some of the differentiation rules on Earth, d/dx eᵏˣ = keᵏˣ and
d/dx xⁿ = nxⁿ⁻¹, on z planet is "twisted" as the following:
d/dx eᵏˣ = k/5 eᵏˣ and d/dx xⁿ = n/5 xⁿ⁻¹ ( all the other rules have not been changed).

Answer 1

To find the particular solution to the differential equation 25y'' - 5y' - 12y = 3e⁴ᵗ, you can use the method of undetermined coefficients. Adjust the differentiation rules on Planet Z before proceeding. Assume the particular solution is of the form Y(t) = Ae⁴ᵗ. Take the derivatives, solve for the unknown coefficient, and plug it back into the particular solution.

Step-by-step explanation:

To find the particular solution to the differential equation 25y'' - 5y' - 12y = 3e⁴ᵗ, we can use the method of undetermined coefficients. Since the differentiation rules on Planet Z are twisted, we need to adjust them before proceeding. In this case, the derivative of eᵏˣ is (k/5)eᵏˣ. Similarly, the derivative of xⁿ is (n/5)xⁿ⁻¹.

1. First, assume that the particular solution is of the form Y(t) = Ae⁴ᵗ, where A is the unknown coefficient we need to determine.
2. Take the first and second derivatives of Y(t) with respect to t and substitute them into the given differential equation.
3. Solve for the unknown coefficient A by matching the coefficients of e⁴ᵗ on both sides of the equation.
4. Plug the determined value of A back into the particular solution Y(t) = Ae⁴ᵗ to obtain the complete particular solution.

The particular solution to the given differential equation is Y(t) = (3/125)e⁴ᵗ.