MATH 202 Exam, Fall 2002

American University of Beirut Math 202 Quiz #1, October 23,2010 1.Solve The DE 𝒅𝒚 𝒅𝒙= 𝒚−𝒙 𝒚+𝒙 2.Find an integrating factor and use it to solve the following IVP (2y2+3x+6)dx + 2xydy =0 , y(1) =1 3.Solve the following IVP x 𝒅𝒚 𝒅𝒙+y =y2f(x) ,y(1)=1 f(x)= 𝒙𝟐, 𝟏 ≤ 𝒙 ≤ 𝟑 𝟐 𝟎, 𝟑 𝟐 < 𝒙 < 𝟐 4.In this problem the indicated function y1(x) is a solution of the given differentail equation. Use reduction of order to find a second non-trivial solution y2(x),and then compute the Wornskain W(y1,y2) xy"+y' = 0; y1=lnx 5.In this problem you may use the fact that the area of an ellipse of equation 𝒖𝟐 𝒑𝟐 + 𝒗𝟐 𝒒𝟐 = 𝟏, is 𝜋pq. Let S be the part of the ellipsoid 𝑥2 𝑎2 + 𝑦2 𝑏2 + 𝑧2 𝑐2 = 1 in the first quadrant .[The "curved traingular surface ABC in the figure]. Let D be the solid , in the first octant, bounded by S and the coordinate planes, x=0,y=0,and z=0. (a) What are the outward unit normals to the four faces OAB,OBC,OCA,and ABC of the solid D? Use the vector field F=i+j+k and the divergence theorem to evalute the surface integral 𝒙 𝒂𝟐 + 𝒚 𝒃𝟐 + 𝒛 𝒄𝟐 ( 𝒙 𝒂𝟐)𝟐 + ( 𝒚 𝒃𝟐)𝟐 + ( 𝒛 𝒄𝟐)𝟐 𝑺 𝒅𝝈 (b) If L is the path made up of the arc AB,followed by the arc BC,followed by the arc CA,evalute the line integral 𝑭. 𝒅𝒓 𝒍