Differential Equations Answers

Consider the following system of differential equations representing a prey and predator

population model

x y

dt

dx

= −

2



x y

dt

dy

= +

i) Identify all the real critical points of the system

ii) Obtain the type and stability of these critical points.

write the ordinary differential equation (1+sin y)dx = (2ycosy-x(secy+tan y))dy

Using the method of undermined coefficients, find the general solution of the differential equation y^iv-2y'''+2y''=3e^-x+2e^-xx+e^-x sin x

Find the solution of the Riccati equation

dy/dx=(2cos^2(x)-sin^2(x)+y^2)/2cos(x); y1(x)=sinx

Find the value of b for which the equation

( ye^(2xy)+x)dx+bxe^(2xy)dy= 0

is exact, and hence solve it for that value of b

Solve: z(p-q) = z^2 + (x+(y^2))

A series RLC circuit with R = 6 ohm, C = 0.02 Farad and L = 0.1 has no applied voltage. Find the subsequent current in the circuit if the initial charge, on the capacitor is q0 and the initial current is zero.

Consider the following first-order ODE formulations

0 0

{ ( )}, ( )

( )

a L n t n t n

dt

dn t

= − =

Associate the physical meaning to the variables {t, n(t)} and the parameters {a, L} so that the

above formulation becomes a mathematical model for population changes.

The population x(t) of a certain city satisfies the logistic law



2

8

10( )

1

100

1

x x

dt

dx

= −

where t is measured in years. Given that the population of the city is 100000 in 1980,

determine the population at any time t >1980 . Also find the population in the year 2000.

Solve the differential equation dy/dx+xy=y^2e^(x^2/2)sinx