Differential Equations Answers

sketch the gradient field for the differential equation dy/dt=t-y² for t=-1......,1

y=-1,.......,1

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

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

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 rumor. After 2days 10 people have heard the rumor

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

The supply S and demand D for a particular commodity satisfy the equations

S(p)=100+p+p^t and D(p)=200-p-p^t

1. Find the equilibrium price p(t) at any time t
2. Find the long range equilibrium price
3. graph the particular solution p(0)=75 and p(0)=25 hence discuss the long term behaviour of the price of this commodity

find a series solution in powers of x of the equation

2x²d²y/dx²+xdy/dx+(x²-1)y=0

find the general solutions for each of the following

1. (3x²+9xy+5y²)dx-(6x²+4xy)dy =0 y(2)=-6
2. x²dy/dx +xy=y³/x

solve

1.d²y/dx²-2dy/dx+5y=e^xtan2x

2.dy/dx+3y=3x²e^-3x

1. solve each of the following

a) (6xy+2y²-5)dx +(3x²+4xy-6)dy=0 y(1)=2

b)x²d²y/dx²+xdy/dx+4y=2xlnx

solve 2xydy+(3x^2+4xy)dx=0