Differential Equations Answers

An object of mass m kg of projected vertically upward from the gravity with initial velocity u ms-1. assumes that the forces acting on the object are the gravitational forces and the retarding force due to the air resistance with a magnitude of alpha modulus velocity where alpha is a positive constant and velocity is the object at time.
1. if the mass of the object is 0.1 kg, the initial velocity is 5 ms-1, alpha is 1.0 kg s-1, and g is 10 ms-1
a) calculate the time taken for the object to reach the highest point
b) estimate the time taken for the object to descend from the highest point in the ground
2. Repeat step 3 with a mass 0.1 kg, and various masses alpha between 0.02 kg s-1to 2.0 kgs-1. tabulate your results in an appropriate table.

Obtain the gradient of the following scalar field:
U(p,q,z)=p^2zcos2q

Using the method of undetermined coefficients, write the trial solution of the equation
y"+2y'+5y=x(e)1/x cos 2x
and hence solve it .

solve the IVP x2y'=cos x - 2xy , y(pi)=0, x>0 .

Find the differential equation of the family of surfaces fi(x+y+z, x2+y2-z2)=0 .
What is the order of this p.d.e ?

The ends A and B of a rod 20cm long have the temperature at 30 degree centigrade and 80 degree centigrade respectively until steady state prevails . The temperature of the ends are changed to 40 degree centigrade to 60 degree centigrade respectively . Find the temperature distribution in the rod at time t .

solve: (D2+2DD'+D'2)z=12xy .

using Jacobi's method find the complete integral of the equation
2u1xz+3u2(y)2+(u2)2u3=0 .

Find the directional derivatives of f(x,y,z)=x2+2y2+3z2 at P(1,1,1) in the direction of a=i+j+k .

Solve the differential equation p2+2py cot x-y2=0 , where p=y' .