Differential Equations Answers

A tightly stretched string with fixed end points x=0 and x=1 is initialy in a position given by y=y0 sin^3(pi.x/l). it is released from rest from the initial position. find the displacement y(x,t)

Do the functions y1(t)=root t and y2(t)=1/t form a fundamental set of solutions of the equation 2t^2 y'' +3t y' -y=0, on the interval 0 less t less ~? justify your answer.

A certain population is known to be growing at a rate given by the logistic equation dx/dt=x(b-ax) show that the minimum rate of growth will occure when the population is equal to half the equilibrium size, that is when the population is b/2a.

Solve the differential Equation
d^2y/dx^2 +3dy/dx -10y=3x^2

show that the function
1) u(x,y)=tan^-1(y/x) is a solution of the two dimensional laplaces equation.
2) u(x,t)=e^-(6t) cos2x is a solution of the one dimensional heat equation.

Show that the function u(x,y)=tan^-1(y/x) is a solution of the two dimensional Laplace's equation.

Solve the following ODE using the power series method:
(x+2)y"+xy'-y=0

X+y=1/9
Y+z=2/9
Z+x=5/9

y=2^x.sin(tanx).dy/dx

Obtain all the first and second order partial derivatives of the function: f (x, y)=x^2 sin y+y^2 cos x