# What is the general solution of the differential equation? # z'''-5z''+25z'-125z=1000 #

# z(x) = e^(5x)+Acos(5x)+Bsin(5x) - 8 #

Presuming we've got:

Complementary Role

The equation homogeneous linked to [A] is

Additionally, the related auxiliary equation is:

Consequently, the homogeneous equation's solution is:

Specific Resolution

For this specific equation [A], the following is a likely solution:

Thus, we arrive at the specific solution:

Overall Resolution

which ultimately results in the GS of [A}

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The general solution of the given differential equation is:

z(t) = c₁e^(5t) + c₂te^(5t) + c₃t^2e^(5t) + 20t^3 + 300t + 400

Where c₁, c₂, and c₃ are arbitrary constants.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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