Differential Equations Answers

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

Verify that the equations i) z =sqrt (2x + a )+ sqrt(2y + b) and ii)z^2+u=2(1+l ^x)(x+ly) are both complete integrals of the PDEz=1/p+1/q . Also show that the complete integral (ii) is the envelope of one parameter sub-system obtained by taking b=-a/l -μ/1+l in the solution (i)
Expert's answer

Interpret the initial value problem
0
0
0
2
2
2
0, (0) , w
q
b q q q
q
=


+ = =
t= dt
d
dt
d
for any physical situation and hence solve the problem

Solve: z( p − q) = z^2 + (x + y^2)

Two kinds of bacteria are found in a sample of tainted food. It is found that the populations size of type 1,\\(N_1\\) and of type 2, \\(N_2\\) satisfy the equations \\(\\frac{dN_1}{dt}=-k_1N_1, N_1(0)=N_{1,0}\\) and \\(\\frac{dN_2}{dt}=-k_2N_2, N_2(0)=N_{2,0}\\) . Then the population sizes equal \\(N_1=N_2\\) at the following time

Find the integral surface of the partial differential equation
(x y) y p (y x) x q (x y ) z 2 2 2 2 − + − = +
through the curve , 0 2 xz = a y =

solve D.E. y''=1+(y' )^2

apply method of variation of parameters to solve D.E.
a)x^2y''+xy'-y=x^2e^x