Differential Equations Answers

Solve True / False by giving reasons for your answers.

a) The integrating factor of the differential equation

xdx + ydy = m(xdy + ydx) is .

1/xy. ?

b) The equation (δ^2 z)/ ( δ^2 x) + x^2 × (( δ^2 z) / ( δ y^2)) = 0

is hyperbolic. ?

c) The solution of the differential equation

( D^3 + D^2 D' - DD'^2 - D'^3 ) is

z = θ1 ( y + x ) + θ2 ( y - x ) + x^2 θ3 (y-x ) ?

d) The Pfaffian differential equation

a^2× y^2 × z^2 dx + b^2 × z^2 × x ^2 dy + c^2 × x ^2 ×y ^2 dz = 0 is Integrable. ?

e) The general solution of the equation x ^2 × y'' + xy'- y = 0,

defined in [0,1] is given

y = c1 × x + c2 × x^-1 ?

Solve (D2- 7DD’+6D’2) z = 0

A pond on a fish farm has a carrying capacity of 1000 fish. The pond was originally stocked

with 100 fish. Let N(t) denote the number of fish in the pond after t months.

a) Set up a logistic differential equation satisfied by N(t), and plot an approximate graph

of a fish population.

b) Find the size of the population of fish with the highest rate of growth. Find this rate

given that the intrinsic rate of growth is 3.

Consider and solve the initial value problem of a new form of life discovered on a distant planet. Outside its habitable zone, the rate of change of population of the life form is governed by the following data: 𝑑𝑦 𝑑𝑥 + 𝑦 = 𝑓(𝑥), 𝑤ℎ𝑒𝑟𝑒 𝑓(𝑥) = { 𝑒 −𝑥 , 0 ≤ 𝑥 < 2 𝑒 𝑥 , 𝑥 ≥ 2 ; 𝑦(0) = 1

Draw a direction plot using given differential equation. Also find the equilibrium

solution and plot.

a) y

′ = −2 + t − y

b) y

′ = 3sint + 1 + y