Doctor of Mathematics

Evaluate ∫∫e^-y/ydxdy from x=0 to x=♾️ (infinity) y=x to y=♾️ Change of order of integration

Evaluate closed line integration 1/z-2w.r t. z along the circle |z-2|=4.

Solve:(D^2+2DD'+D'^2)z=e^2x+3y. (Linear Partial Differential Equation with constant coefficients)

Solve (D^2-3DD'+2D'^2)z=e^2x-y

To prove curl(AxB)=

Solve: p^2+2pycotx=y^2, where p stand for dy/dx

cosθ+cos3θ+cos5θ+........ upto n terms ,Sum to n terms (summation of series)

Trace thw curve y^2=x.(x-1)^2 and find the area of the loop.

Construct a farward difference table from the data x:0 1 2 3 4 f(x):

Solve:x^2y"-2x(1+x)y'+2(1+x)y=x^3. or Solve:x^2d^2y/dx^2-2x(1+x)dy/dx+2(1+x)y=x^3

Find the orthogonal trajectories of the family of parabolas y^2=4ax.

Rule 1:For solving the equation dx/P=dy/Q=dz/R.Example: xdx/y^2z=dyxz=dz/y^2

Prove that the line x + y - 3= 0 touches the conic 2x^2+ 4xy + 3y^2 - 4x - 10 y +7 = 0.

Prove that the lines x-1/2=y-2/3=z-3/4 and 4x-3y+1=0=5x-3y+2 are coplanar.Three dimensional (3D).

Solve:(D^2-5DD'+6D'^2)z=e^x+2y P.D.E. with constant coefficients

Prove that for a curve given by r^2=a^2.cos2θ,we have ρ=a^2/3r. (Radius of curvature)

Find the missing values in the following table:x:1 2 3 4 5.y:7 _2740

Evaluate ∫∫e^-y/ydxdy from x=0 to x=♾️ (infinity) y=x to y=♾️ Change of order of integration

(2xz-yz)fx+(2yz-xz)dy-(x^2-xy+y^2)dz=0. Total Differential Equations

Rule 3:For solving the equation dx/P=dy/Q=dz/R.Example:dx/(mz-ny)=dy/(nx-lz)=dz/(ly-mx).

Solve:2xz-px^2-2qxy+pq=0 (Partial Differential Equation of the First order)

Solve:(2x^2+2xy+2xz^2+1)dx+dy+2zdz=0 (Total differential equations)

(D^2+D-2)y=x+sinx. (Linear differential equation with constant coefficients)

Prove that Jn(x)=1/π∫cos(nθ-xsinθ)dθ,n=0,1,2,....(Integral Representation of Bessel Function)

Find the envelope of the straight lines xcosα+ysinα=lsinαcosα.

dx/y+z=dy/z+x=dz/x+y Simultaneous Linear differential equations

Solve:L-1{4p+5/(p-4)^2(p+3)}

A particle is describing a plane curve.If the tangential and normal accelerations are each constant

(D^2-DD'-2D'^2)z=(2x^2+xy-y^2)sinxy-cosxy

(D^2-4D'^2)z=4x/y^2-y/x^2 (Linear Partial Differential Equations with constant coefficients)

Prove that the line x + y - 3= 0 touches the conic x^2+ 4xy + 3y^2 - 4x - 10 y +7 = 0.

Expand logcosx in power of x and verify the coefficient of x^4 algebraically by the use of well kno

Solve:(x-y-2)dx+(x-2y-3)dy=0

Solve: (D^2+3DD'+2D'^2)z=x+y

Solve:(D^2-5DD'+4D'^2)z=sin(4x+y).

If A,B are skew-symmetric matrices of the same order ,then prove A+B also skew-symmetric matrix.