Matrices, Differentiation and Integration (Math Problem Sample)

Provide solution to the following questions:
1. Evaluate the following:
 
By employing ILATE Rule:
uv dx=uv dx- dudxv dxdx
Let y = x sin3x dx
Therefore, xxsin 3xdx -ddx(x) . (sin3x dx) dx
= x(-cos3x)3 - 1. -cos3x3 dx
= -xcos3x3 + 13 cos3x dx
= -xcos3x3 +sin3x9 + C
2. If, then for what value of α is A an identity matrix?
Comparing the given matrix A with an identity matrix 1001 ,
We find that cos∝ = 1, - sin∝ = 0 , -cos∝ = 1 and sin∝ = 0
For cos∝ = 1, ∝must be 0° sincecos0° is=1.
For sin∝ = 0, ∝must be 0° sincesin0° is 0
Therefore, the required value of ∝must be 0° for it to be an identity matrix.
3.  The line y = mx + 1 is a tangent to the curve y2 = 4x .Find the value of m.
The tangent and the line passing in the middle of the curve are perpendicular.
Differentiating the equation of the curve we have:
2y dydx = 4
Therefore, dydx = 2y
This means the slope of the curve will be m = 2y and therefore y= 2m
When y= 2m , x = y24 = 4m2 ÷ 4 = 1m2
Hence x = 1m2 and y = 2m will lie on the same line at the point of tangent.
y = mx + 1
2m = m . 1m2 + 1
2m = 1m + 1
2m - 1m = 1
1m = 1
m = 1
4. Solve the following differential equation: (x2 − y2) dx + 2 xy dy = 0.
This is a form of homogenous differential equation;
dydx= - x2-y22xy
Making y = ux , we obtain u+ xdudx=dydx
Utilizing the form u+ xdudx , we have -x2 x2-u22x2u = - 1-u22u
xdudx= -1-u22u+u
xdudx= -1+u22u
2u1+u2 du= -dxx
Finding the integration of both sides gives;
2u1+u2 = -dxx
Introducing logarithms on both sides gives, log1+u2= - logx+logC
logx2+y2x2x=logC
x2+y2x=C
x2+y2=Cx
When y=1, x = 1 and hence 1+1=C;C=2
Therefore x2+y2=2x
=2x
5. Solve system of linear equations, using matrix method.
x − y +2z =73x +4y −5z =−52x − y + 3z = 12
Making the equation in the matrix form Ax=b
1-1234-52-13 xyz = 7-512
Let A