(1) Find L{sin(2t)}. (5%) (Write down the detailed process)

(1) Let u1 = ＜3, 1＞, u2= ＜1, 1＞. Transform them into an orthonormal basis. (5%)

(2) Let u1 = ＜1, 1, 1＞, u2 = ＜1, 2, 2＞, u3 = ＜1, 1, 0＞ . Transform them into an orthonormal basis. (5%)

3.Let . Find a symmetric matrix B and a skew-symmetric matrix C, such that B+C=4. 1 0] (5%)

4.Solve +y=x, y(0)=3.(10%)

(2) Solve +3y= 13sin(2t), y(0)=6.(15%)

6.Please write down the Bessel's differential equation. (7%)

7. Find y when y"x2 + (x2 -81)y = -xy'.(10%)

8.Find the eigenfunction of the following equation: y"+ ky =0, y(0)-y(L)-0.(15%)

9. Find the Fourier series in complex form for the periodic function f(t) with period 2, .(18%) 

7. Bigly or Big League? "Did he say bigly?" This question had been asked repeatedly on social media soon after the then-candidate Donald Trump was reported to use the word during the first US presidential debate. Many wondered whether bigly was really an English word or a "Trumpism" presumably derived from big and the adverb suffix -ly. However, based on acoustic analyses, some phoneticians argued that what Mr. Trump actually said was big league but the final /g/ in his league was just difficult for the untrained ear to detect. Figure 2 shows the waveform and spectrogram of Trump's production of what appears to be Figure 1 Donald Trump, the to-be ambiguous between bigly and big league. Look at the figure. president of the most powerful What are the acoustic cues to final /g in league? Name at least  TWO of them.