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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