Differential Equations Answers

Solve the differential equation
Suppose the temperature of a body when discovered is 85° F. Two hours later, the temperature is 74°F and the room temperature is 68°F. Find the time when the body was discovered after death (assume the body temperature to be 98.6°F at the time of death.)

Solve the differential equation
1.sin^-1(dy/dx)=(x+y)
2.(1+y^2)dx=(tan^-1y-x)dy
3.(D-1)^2(D^2+1)^2y=sin^2(x/2)+e^x+x
4.2x^2y(d^2y/dx^2)+4y^2=x^2(dy/dx)^2+2xy(dy/dx)

Obtain a solution of the wave equation
∂^2u(x,t)/∂t^2=16(∂^2u(x,t)/∂x^2)

for 0≤ x ≤ π and t > ,0 and the following boundary and initial conditions:
u(0,t)=u(π,t)=0,
u(x,0)=x(π-x) and ∂u(x,0)/∂t=0

Reduce the following PDE to a set of three ODEs by the method of separation of
variables (1/r)(∂/∂r)(r(∂V/∂r)+(1/r^2)(∂^2V/∂θ^2)+(∂^2V/∂z^2)+(k^2)V=0

Obtain a solution of the wave equation
∂^2u(x,t)/∂t^2=16(∂^2u(x,t)/∂x^2)

for 0≤ x ≤ π and t > ,0 and the following boundary and initial conditions:
u(0,t)=u(π,t)=0,
u(x,0)=x(π-x) and ∂u(x,0)/∂t=0

Obtain the Fourier series expansion for the following periodic function which has a
period of π: f(x)={(4/π) for 0<x<π/2

obtain all the first and second order PD of the fuction f(x,y)=(x^2)Siny+(y^2)cosx

show that function u(x,t)=(exp^-(mu)t)sin x is a solution of of one dimensional heat equation

A block of mass 1 kg is attached to a spring of force constant k = 25/4 N/m. It is pulled x = 0.3 m from its equilibrium position and released from rest. This spring‐block apparatus is submerged in a viscous fluid medium which exerts a damping force of – 4 v (where v is the instantaneous velocity of the block). Determine of the position x(t)
of the block at time t.