4 5(10pt). Use Laplace transform to solve the equation: (3) + x = 0, x(0) = 0, '(0) = 0, "(0) = 1. 6(10pt). Find the inverse Laplace transform of F(s) = ( using Theorem 1 on page 298. And we recall from Example 5 of Section 4.2 that the Laplace transform of sing is indeed s/(s²+1)². Theorem 1 is proved at the end of this section. THEOREM 1 The Convolution Property Suppose that f(f) and g(1) are piecewise continuous for 10 and that [f(t)\ and g(t) are bounded by Me as t+oo. Then the Laplace transform of the convolution f(t)* g(t) exists for s > c; moreover, A L{f(t)g(1)) = L{f(t)} L{g(t)} (4) and L{F(s) G(s)} = f(t) * g(t). (5) Thus we can find the inverse transform of the product F(s) G(s), provided that we can evaluate the integral L¹{F(s) G(s)}= f(t)g(tt) dr. (5') Example 2 illustrates the fact that convolution often provides a convenient alternative to the use of partial fractions for finding inverse transforms. With f(t) = sin 21 and g(t) = e', convolution violdo