Engineering Mathematics1 : Midterm Exam

October 24, 2007

1. (20 points) Consider the following matrix form of a linear system

[eqn: Ax`=`b]

where [eqn: A` IN `R ^{n TIMES n} ,`x` IN R ^{n} ,`b` IN R ^{n} .`]Let [eqn: hat{A}] be the [eqn: n TIMES n] matrix consisting of the first n column vectors of the row echelon form obtained at the end of Gauss elimination. Also, let [eqn: hat{b}]be the last column vector of the row echelon form. Then, prove by use of elementary matrices that if [eqn: hat{x}] is the solution of [eqn: hat{A} hat{x} = hat{b} `], then [eqn: hat{x} =x]

2. (20 points) Find an eigenbasis of the following matrix and diagonalize the matrix(Show the details)

[eqn: {bmatrix{-2.5&`````-3&```````3#-4.5&````-4&``````6#-6&````-6&``````8}}]

3. (20 points)  Test for exactness of the following equations. If exact, solve the equations. If not, use an integrating factor to solve equations.

a) [eqn: 2y ^{-1} cos2xdx=y ^{-2} sin2xdy]

b) [eqn: 2xydy=(x ^{2} +y ^{2} )dx]

4.  (20 points)  Consider 1st order nonhomogeneous linear ODEs.

a) Find the general solution of the following equation based on mathematically logical development

[eqn: y ^{'} +p(x)y=r(x)]

b)  Find the particular solution of the following equation using the formula obtained in 4(a)

[eqn: y ^{'} -(1+3x ^{-1} )y=x+2], [eqn: y(1)=e-1]

5.  (20 points) Consider Euler-Cauchy equations.

a) Prove that the solution of [eqn: x ^{2} y ^{''} +axy ^{'} +by=0] is [eqn: y= LEFT ( c _{1} +c _{2} lnx RIGHT ) x ^{LEFT ( 1-a RIGHT ) /2}] when [eqn: LEFT ( 1-a RIGHT ) ^{2} -4b=0]. where, [eqn: c _{1} ,`c _{2}] is arbitrary constant.

b)  Find a general solution of the following equation.

[eqn: x ^{2} y ^{'' } -xy ^{prime } +2y=0]