課程名稱︰應用數學二 課程性質︰物理系必帶 課程教師︰李慶德 開課學院：理學院 開課系所︰物理學系 考試日期（年月日）︰2018/11/14 考試時限（分鐘）：180分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Do any 5 of the exam problems given below! 1.Consider the differential equations (a)(e^x siny-2y sinx)dx+(e^x cosy+2cosx)dy=0; (b)ydx+(2xy-e^(-2y))dy=0 Solve the equations immediately if they are exact equations. Otherwise, try to solve the equations by first finding an integrating factor that depends only on x (or only on y). 2.Solve the differential equations: (a) y'=1/(e^(-y^2/2)-xy) (b) y'=x^2+y^2+2xy+4x+4y+4 3.Often a differential equation with variable coefficients such as y''+p(t)y'+q(t)y=0 can be put in a more suitable form for finding a solution by making a change of independent and/or dependent variables. (a)Let x=u(t)=∫[αq(t)]^(1/2)dt be the new independent variable in which α is a constant chosen such that αq(t)>0. Show that if q'(t)+2p(t)q(t) -------------------- [q(t)] ^(3/2) is a constant, then the differential equation above can be transformed into an equation with constant coefficients by the change of variables t→x=u(t). (b)Solve the differential equation y''+ty'+e^(-t^2)y=0 through a suitable change of independent variables. 4.Find the solutions of the following differential equations: (a) y''+y=tan(x), y(0)=1, y'(0)=0. (b) y^(iv)+2y''+y=2cos(x)-sin(x), where y^(iv)≡d^4y/dx^4 Discuss also the range of validity for the unique solution of (a). 5.Solve the differential equation xy''-2(x+1)y'+(x+2)y=e^x with the initial conditions y(1)=y'(1)=0, given that the solutions of the homogeneous equation are of the type g(x)e^x. 6.Consider the differential equation L[y]≡y''-y'-2y=1/(e^x+1) (a)Factorizing L[y]=(D+1)(D-2)y, where D≡d/dx, solve the differential equation by using the relations (D+1)u(x)=1/(e^x+1), (D-2)y(x)=u(x). (b)Find the general solution of the differential equation by the method of variation of parameters. 7.Consider the differential equation y'''+y''tanx+(3+2tan^2x)y'+(3tanx+2tan^3x)y=sinx. (a)Show that the differential equation can be factorized into the following form: (D^2+1)(D+tanx)y=sinx where D≡d/dx is the differentiation operator. (b)Use the result of (a) to find the general solution of the differential equation. --

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