Differential Equations Answers

Let x

α

y

β be an integrating factor of x(4ydx + 2xdy) + y

3

(3ydx + 5xdy) = 0. Find α, β and

the solution of given differential equation.

A string is stretched and fastened to two points at a distance 0

l

0 apart. Motion is started by

displacing the string in the form y = k sin πx

l

from which it is released at time t = 0. Show

that the displacement of any point on the string at a distance x from one end at time t is

given by k sin πx

l

cos πax

l

Solve the ordinary dierential equation

y'' + 2y' + 5y = 4e^-x + 17 sin 2x:

Find the general solutions of the following equations.

LINEAR DE

1. dx/dy - 4x/y = y⁵
2. dy/dx + y/x = cos x + (sinx/x)
3. dx/dy + x(2-y)/y = 1/y

BERNOULLI'S EQUATION

1. dy/dx - 3y/2x = y³/2x³
2. dy/dx - y/2x = -x²/y
3. dy/dx+ y = 2xy²e^x

The yield y(t) (in bushes)per acre of a corn crop satisfies the equation dy/dt+y=100+e^-t

if y(0)=0 find y at any time t

2.The population N(t) of a species of micro-organism in a laboratory setting at any time t is established to vary under the influence of a certain chemical at a rate given by dN/dt=t-2e^t show that N(t)=t-2e^t+c.hence if N(0)=400 determine the population when t=5

3.A single individual starts a rumour in a community of 200 people. the rumour spread at a rate proportional to the number of people who have not yet heard the rumour. After 2days 10 people have heard the rumour

1. how many people will have heard the rumour after 5days
2. how long will it take for the rumour to spread to 100 people

The population N(t) of a species of micro-organism in a laboratory setting at any time t is established to vary under the influence of a certain chemical at a rate given by dN/dt=t-2e^t show that N(t)=t-2e^t+c.hence if N(0)=400 determine the population when t=5

(D²-3DD'+2D'²)z=(2+4x)e^x+2y.