课程名称︰工程数学 课程性质︰生机系必修 课程教师︰颜炳郎 开课学院：生农学院 开课系所︰生机系 考试日期（年月日）︰108.11.05 考试时限（分钟）：180 试题 : 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). --

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