課程名稱︰工程數學
課程性質︰生機系必修
課程教師︰顏炳郎
開課學院：生農學院
開課系所︰生機系
考試日期（年月日）︰108.11.05
考試時限（分鐘）：180
試題 :
Solve the following differential equations: 10% for each problem
1. 2cos(x+y) - 2xsin(x+y) - 2xsin(x+y)y' = 0
2. xy' = 2xcos(y/x) + y, y(0) = 1
3. x^2y' = xy + 2y^2
4. y' + 4xy = e^{-2x^2}, y(0) = -4
5. y'' + y' + y =secx
6. x^2y'' + xy' 0 4y = lnxcos(lnx)
7. y'' - 4y = e^{2x}cosx, y(0) = 0, y'(0) = 1
8. y'''' + 6y''' + 18y'' + 24y' + 16y = x^2 + e^{-x}sinx
9. y'' - x^2y + y = 0
10. y'' - y' + y/x = 0
Bonus questions: 10% for each problem
1. x'' + ω^2x = f(t), please use the variation of parameter method to prove
t
the general solution is x = Asinωt + Bcosωt + (∫f(τ)sinω(t-τ)dτ)/ω
0
2. y' = a(x) + b(x)y + c(x)y^2 is known as Riccati's equation and is of speci-
al importance in the study of optimal control.
(a) Show that if y = Y(x) is any particular solution of the Riccati's equation
, then v = 1 / (y-Y(x)) satisfies a linear differential eqution of first order
(b) Find the general solution of y' = 1 + (y-x)^2 (Hint: use the particular
solution y = x).