110年 - 110 國立中山大學碩士暨碩士專班招生考試_機電系碩士班/乙、丙組:工程數學#104304

1. Find the general solution of the first-order ODE: (2x³ + y³)dx -3xy²dy = 0.(10%)

(a) Represent it as a vector equation y' = Ay + g. (5%)

(b) Find the eigenvalues and eigenvectors of A. (5%)

(c) Find the general solution of the system. (5%)

(d) Find the particular solution. (5%)

(a) Set up the model. (5%)

(b) Find I₁(s) and I₂(s). (10%)

(c) Find i₁(t) and i₂(t).(5%)

(a)(3A -2B)^T(5%)

(b)(3A-2B)^Ta^T(5%)

5.Find an cigentagi and pecrform diagonalization on the folowing matri. (10%)

6. Let a = [4, 7, 0], b= [3,-1, 5], c=[-6,2,0], and d = [1,-2, 8]. Find the angle between a and c. Between b and d. (10%)

7. Calculate this line integral by Stokes's theorem for the given F and C. Assume the Cartesian coordinates to be right-handed and the z-component of the surface normal to be nonnegative. F= [0,z³,0], C the boundary curve of the cylinder x² + y²= 1, x≧ 0,y ≧ 0， 0 ≦z≦1 (10%)

8. Solve y''+ ω²y = r(t), where -periodic and r(t)=3t²(-π＜t＜π).(10%)

110年 - 110 國立中山大學碩士暨碩士專班招生考試_機電系碩士班/乙、丙組:工程數學#104304

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