课程名称︰ 工程数学 课程性质︰ 系定必修 课程教师︰ 邱文英 开课学院： 理学院 开课系所︰ 化工系 考试日期（年月日）︰ 97/11/5 考试时限（分钟）： 100分钟 是否需发放奖励金： 是 试题 : 1.( 8%) Existence (a) Dose the ODE y'^(2)=-1 have a "real solution" ? (b) Dose the ODE │y'│+││y│= 0 have a "general solution"? 2.(48%) Find a general solution 1 (a) xy'= ── y^(2)+y 2 (b) (e^(x+y)+ye^(y))dx + (xe^(y)-1)dy = 0 (c) (x^(2)+y^(2))dx - 2xydy = 0 (d) y'+(x+1)y=e^(x^(2))y^(3) (e) x^(2)y''+xy'-4y=0 1 (f) y''+9y=cosx+──cos3x 3 (g) y" - 2y + y =e^(x)sinx 1 3 (h) x^(2)y" + xy' - ──y = ── +3x 4 x 3.( 8%) If y1 and y2 are solution of y1' + py1 = r1 and y2' + py2 = r2 respectively, what can you say about the sum y1+y2 ? 4.( 8%) Find the solution of 2(y')^(2) - (2y^(2) + x)y' + xy^(2) = 0, How could you find a singular solution from general solution? 5.(10%) Find the general solution for x(t) and y(t) ┌ dx │ ── - 3x -6y = t^(2) │ dt │ │ dy dx │ ── + ── - 3y = e' └ dt dt 6.(10%) Prove that 1 (a) y = ───e^(ax) Φ(D) e^(ax) = ─── if Φ(a)≠ 0 Φ(a) 1 (b) y = ───e^(ax) Φ(D) xe^(ax) dΦ = ─── if Φ(a) = 0, ───│ ≠ 0 dΦ dD D = a ──│ dD D = a where Φ(D) = anD^(n) + a(n-1)D^(n-1) + ...... + a1D + ao, and an, a(n-1)...a & a are constants. 7.( 8%) (Spring) Find the frequency of vibration of a ball of mass m=3kg on a spring of modulus (a) k1=27nt/m (b) k2=75nt/m (c) on there springs in parallel --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 140.112.217.11