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How to use Laplace theorem?
Laplace theorem is an important tool in mathematical analysis, which is used to solve a special kind of differential equation problems. Its core idea is to transform the higher derivative of a function into its Laplace transform.

Specific use:

Specifically, if the Laplace transform of function f(t) is F(s), Laplace theorem gives the relationship between Laplace transform of n-order derivative of function f(t) and F(s). According to Laplace theorem, for any positive integer n, the following equation holds:

L{f'(t)} = sF(s) - f(0)? (first derivative).

L{f''(t)} = s^2F(s) - sf(0) - f'(0)? (second derivative).

...

l{f^n(t)} = s^nf(s)-s^(n- 1)f(0)-s^(n-2)f'(0)-...- f^(n- 1)(0)? (n-order derivative).

Where L{f(t)} represents the Laplace transform of function f(t), f'(t) represents the first derivative of f(t), f' (t) represents the second derivative of f(t), and f n (t) represents the n derivative of f(t).

Solution:

Through Laplace theorem, we can turn the problem of solving differential equations into the problem of solving algebraic equations. The concrete steps are as follows: firstly, the differential equation is laplace transformed to obtain the algebraic equation about F(s); Then solve f (s); Finally, the solution of the original function f(t) is obtained by inverse Laplace transform.

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