課程名稱︰工程數學一
課程性質︰必修
課程教師︰李克強
開課學院：工學院
開課系所︰化學工程學系
考試日期（年月日）︰2014/11/21
考試時限（分鐘）：110（後來延長至130min)
試題 :
Find the solutions of the following ODEs
(一) (5% each)
dy 1
1. ---- + --- = 3cos(2x), x > 0
dx x
dy
2. ---- = y - (x^2)(y^3)
dx
dy x^2
3. ---- = ---------
dx 1 + y^2
4. (ye^2xy + x)dx +x(e^2xy)dy = 0
5. (x^2)(y^3) + x(1+y^2)y' = 0 (Hint: Try the Integrating Factor μ=1/xy^3)
dy x^2 + 3y^2
6. ---- = ------------
dx 2xy
(二) (6% each)
1. (1+x^2)y" + 2xy' +3x^(-2) = 0, x > 0
2. y" - 9y' + 9y = 0
3. y" + y' + y = sin^2 x
4. y" - y' - 2y = exp(2x) + cos x - 4x
5. (x^2)y" + xy' - y = 1 + x
(三)
d^2 y
1.(6%) ------- + y = δ(t-1) + μ(t-2), y(0) = 0, y'(0) = 0
dt^2
d^2 y
2.(6%) ------- - y = δ(t-1)sin t , y(0) = 0, y'(0) = 1
dt^2
d^2 y
3.(10%) ------- + y = f(t) , y(0) = 1, y'(0) = 0
dt^2
┌ 1, 0 ≦ t < π
where f(t) = ┤ and f(t+2π)=f(t), i.e., period=2π
└ 0, π≦ t < 2π
dx
┌ ---- + x = exp(t)
4.(15%)│ dy y(0) = 0, x(0) = 1
│ dx
└ y - 2---- = 1
dt
5.(3%) Find Laplace Transform of the function:
f(t) = ∫f(t-τ)^2 cos(2τ)dτ, τ[0,t](上下限範圍)
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