Differential Equations Answers

The solution of the initial value problem dy/dx=-x/y, y(4)=3 is

Newtons laws of cooling proposes that the rate of change of temperature is proportional to the temperature difference to the ambient (room) temperature. And can be modelled using the equation: dT/dt = -k (T-Ta)

It can also be written as dT/T-Ta = -k dt

Where:
T = Temperature of material
Ta = Ambient (room) temperature
k = A cooling constant

a) integrate both sides of the equation and show that the temperature difference is given by:

(T-Ta) = CoE^-kt

(Co is a constant for this problem)

B) calculate Co if the initial temperature is 70 degrees C and Ta = 20 degrees C?

solve the following differential equations
1- (y-2)dx-(x-y-1)dy=0
2- (x-4y-9)dx+(4x+y-2)dy=0

Find the complete integral using Charpit Method : 2x(q²z²+1) = pz

d^2y/d^2 = 1/x(x+1) + cosec^2x

Find T(x,t) in a laterally insulated 2 m-long rod if k=10-4 m2/s and T(x,0)=100(2x-x2), T(0,t)=0=T(2,t).

Compute the n-th differential coefficient of \\(y=x\\log_{e}x\\)\n

by Charpits AE sobylve z=p^2x+q^2y

all question of power series
Q.1 For the following differential equation, discuss about Ordinary point, Singular point,
Regular Singular point, Irregular Singular point.
1) (

consider the system:
dx/dt = x^2+y
dy/dt = x^2*y^2
Show that, for the solution (x(t),y(t)) with initial ocndition (x(0),y(0)) = (0,1), there is a time t* such that x(t)--> infinity as t--> t*. In other words the solutions blows up in finite time.