Differential Equations Answers

The taut string of length 50 cm fastened at both ends, is distributed from its position of equilibrium by imparting to each of its points an initial velocity of magnitude kx for 0<x<50. Find the displacement function y(x, t).

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