Differential Equations Answers

y"+(x-1)y'+y=0

x(y − z)p + y(z − x)q = z(x − y)

Solve and find particular solution of the first order non-linear differential equation of Bernoulli’s type with given condition.

(dy/dx)+(xy/1-x² )=xy^1/2 ; y(0)=1

Solve and find particular solution of the first order non-linear differential equation of Bernoulli’s type with given condition.

.  y”  -  y’ -   2y  =  1 – 2x - 9e^-x        

What is the solution for homogeneous equation (2x+y)²dx=xydy

The differential equations

dS/dt=−βSI + λS

dI/dt= βSI + γI

model a disease spread by contact, where S is the number of susceptibles, I is the number 

of infectives, β is the contact rate, γ is the removal rate and λ is the birth rate of 

susceptibles. 

 (i) Identify which term in the RHS of each differential equation arises from the birth of 

susceptibles. 

 (ii) Discuss the model given by the above two differential equations.

A model corresponding to the cooperative interaction between two species x and y

is given by

dx/dt=(4-2x+y)x

dy/dt=(4+x-2y)y

Find all the equilibrium points of the system and discuss the stability of the system at

these points.

In a population of lions, the proportionate death rate is 0.55 per year and the

proportionate birth rate is 0.45 per year. Formulate a model of the population. Solve

the model and discuss its long term behavior. Also, find the equilibrium point of the

model.

A park has a stable population of birds. Prior to this situation, the birds’ population

increased from an initial low level. When the population of birds was 1000, the

proportionate birth rate was 40% per year and the proportionate death rate was 5% per

year. When the population was 3,000, the proportionate birth rate was 30% and the

proportionate death rate was 10%. Consider the population model under the following

assumptions:

(i) There is no migration and no exploitation.

(ii) The proportionate birth rate is a decreasing linear function of the population.

(iii) The proportionate death rate is an increasing linear function of the population.

Show that

The population grows according to the logistic model.

Find the stable population size.

If the shooting of birds is allowed at the rate of 15% of the population per year, find the

new equilibrium population.