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Write the ordinary differential equation ydx+(xy+x-3y)dy=0 in the linear form , and hence find its solution.
Solve dy/DX +xy= y^2e^x^2/2sin x
The pde uxx+ x^2 uxy -(x^2/2+1/4) uyy = 0 is hyperbolic in the entire xy-plane.true or false why?
The solution of the pde dz/dx +dz/dy = z^2 is z = -[y+f(x-y)] true or false why?
The normal form of the differential equation y``-4xy``+(4x^2-1)y = -3e^x^2 sin2x is d^2v/dx^2+v =-3sin2x, where v= ye^-x^2. True or false why?
The initial value problem dy/dx = x^2+y^2, y(0) = 0 has a unique solution in some interval of the form -h<x<h. True or false ,why?
The orthogonal trajectories of all the parabolas with vertices at the origin and foci on the x-axis is x^2+2y^2= c^2. True or false ,why?
State whether the following statements are true or false. Justify your answer. a) The level curves of x/yz=are hyperbolas. b) 2222)0,0()y,x(yxyxlim+−→does not exist. c) The set }0z,0y,0x|)z,y,x{(<>>is a domain in 3R. d) The work done by the Force )x,y()y,x(F−=in moving a particle along the boundary of the ellipse 36y4x922=−+is 6. e) The function xyze)z,y,x(f=is integrable over ].1,0[]1,0[]1,0
Find the directional derivative of f(x,y) = {2 xy/x^2+y^2, (x,y) #(0,0) and 0, (x,y) =(0,0) at (0,0) in the direction of ( i) θ=π/2 (ii) θ= π/4
If z= e^x siny ,where x= st^2 and y= s^2 t, find ∂z/∂s and ∂z/∂t.
