Differential Equations Answers

Solve the following ODE using the Frobenius method:
d^2y/dx^2+(1/2x)(dy/dx)+y

Determine all the first and second order partial derivatives for the function:
u(x, y)=x^2sin(y)+y^2cos(x)

Show that
u=sin(wx)e^(-w^2c^2t)
is a solution of the one-dimensional heat equation.
note : here w stands for omega

Please solve the initial value problem:

xu_x + (1+y)u_y = x(1+y) + xu

u(-1,y) = $(y), where $(y) is a given function such that $ is an element of C'(R). Please explain why this condition on $ must be satisfied.

Solve the initial value problem, and be sure to check the non-characteristic condition first:

U_x + U_y = U^2

U(x,0) = h(x) - given function.

Solve the following IVP, and check the non-characteristic condition first: UU_x + yU_y = x U = 2s on a characteristic curve gamma: x = s, y = s, s element of R.

2. In a factory turning out razor blade, there is a small chance of 1/500 for any blade to be defective. The blades are supplied in a packet of 10. Use Poisson distribution to calculate the approximate number of packets containing blades with no defective, one defective, two defectives and three defectives in a consignment of 10,000 packets.

Solve the following IVP, and check the non-characteristic condition first:

UU_x + yU_y = x
U = 2s on a characteristic curve gamma: x = s, y = s, s element of R.

Solve the following Ordinary Differential Equation
y''+4y=2cosXcos3X

2. In a factory turning out razor blade, there is a small chance of 1/500 for any blade to be defective. The blades are supplied in a packet of 10. Use Poisson distribution to calculate the approximate number of packets containing blades with no defective, one defective, two defectives and three defectives in a consignment of 10,000 packetsab