Differential Equations Answers

Find the z – transform of the function e−2tcos3t

Form the differential equation by eliminating f and ϕ from z = xf(yx)+yϕ(x)

A string is stretched between two fixed points at a distance 2l cm and the points of the string are given initial velocities (v) is given by v(x, 0) = {kxl, 0<x<lk(2l−x)l, l<x<2l, x being the distance from an end point. Find the displacement of the string at any time.

The population of a certain country has grown at a rate proportional to the number of people in the country. At present, the country has 80 million inhabitants. Ten years ago it had 70 million. Assuming that this trend continues:

a. Find an expression for the approximate number of people living in the country at any time t (taking t=0 to be the present time)
b. Find the approximate number of people who will inhabit the country at the end of the next ten-year period.

Uranium disintegrates at a rate3 proportional to the amount present at any instant. If m1 and m2 grams of uranium are present at time t1 and t2 , respectively, show that half life of uranium is (t1-t2)In2/In(m1/m2)

Solve the differential equation:
(D^4-a^4)y=x^4+sinbx, where a and b are constants and D=d/dx

(2xy^2 - 2y)dx + (3x^2y - 4x)dy = 0

Apply Charpit's method to solve the following differential equation:
p(1+q^2)=q(z-a)

Using the method of product solution,solve ∂u/∂x= 2∂u/∂x+u where u(x,0)=6e^-3x

Solve the differential equation
(x-y^2)dx+2xy=0