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Sma2110:Real Analysis Ii Question Paper
Sma2110:Real Analysis Ii
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE (30 MARKS)
a) Define a partial differential equation and give an example. (2 Marks) Hence eliminate the arbitrary constants a and b to find the p.d.e arising from ??2 + ??2 + 2 = 1. (5 Marks) b) Given the tangent planes at the point = 0, 17-1 2 , 17-1 2 of the surface ??2 + 2 + 2 = 4 ???? = ??2 + 2. Find the normal vectors and the equation of the tangent generated by the intersection of the planes at this point. Find the curve generated by the intersection of both surfaces. (7 Marks) c) Obtain the general solution, hence the particular solution of the equation - ?? + ?? - = - ??. (5 Marks) d) Find the orthogonal trajectories on the sphere ??2 + 2 + 2 = 2of its intersections with the paraboloids ?? = ?? where c is a parameter. (6 Marks)
QUESTION TWO (20 MARKS)
a) Find the integral curves of the equations ???? ??(-)
=
?? (-??)
=
?? (??-)
. (6 Marks)
b) Consider a curve which is the intersection of the surfaces ??,, = 0 ???? ??,, = 0. Prove that ????,??,?? ? ??(,) ??(,) , ??(,) ??(,??) , ??(,) ??(??,)
Hence show that the direction cosines of the tangent at the point P(x,y,z) to conic.
2
??2 + ??2 + ?? = 1,?? + + = 1 are proportional to (?? - ??,?? - ??,?? - ??) . (14 Marks)
QUESTION THREE (20 MARKS)
a) Find the general solution hence the particular solution of the d.e. ???? + = 3. (6 Marks) b) Obtain the complete hence the singular solution of the non-linear p.d.e. = ???? + + ??2 + ?? + 2. (7 Marks) c) Solve the equation ???? + = (7 Marks)
QUESTION FOUR (20 MARKS)
a) Show that a necessary condition for the Pfaffian differential equation ???? + ?? + ?? = 0 to be integrable is that . × = 0 where E is a vector Assume twice differentiable. (8 Marks) b) Verify that the differential equation ???? + ???? + 2?? = 0 is integrable and find its solution. (6 Marks) c) Find the integral surface of the equation ?? 2 + ?? - ??2 + = ??2 - 2 which contains the straight line ?? + = 0 ???? = 1). (6 Marks)
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