Differential Equations Answers

Find the general solution of each of the following

i) (2xsiny+y³e^x)dx +(x²cosy+3y²e^x)dy=0

ii) (ysec²x+secxtanx)dx + (tanx+2y)dy=0

iii)(ye^x+2e^x+y²)dx+(e^x+2xy)dy=0 y(0)=6

iv)(2xcosy+3x²y)dx+(x³-x²siny-y)dy=0 y(1)=3

Solve the following differential equation (3 marks)

d

2

y

dx2

− 6

dy

dx + 9y = x

2

e

3x

using the method of undetermined coefficients.

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