Explore BrainMass
Share

10 Differential Equation Questions

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Use separation of variables to solve to the ODE:

N ̇=〖{k〗_+-k_- ln⁡〖(N)}N〗;k_+,k_- and t>0;N(0)= N_0>0.
Hint: Use u = ln(N) for u-substitution.

Substitute your solution into the differential equation and show that it is in fact the solution.

Evaluate your solution for N(t) as t→∞ .

For what values of N will N ̇=0?

For what values of N_0 will
N ̇(0)>0
b) N ̇(0)<0

Considering your answers to questions 3), 4) and 5), describe how N and N ̇ change from t=0 to t →∞ for cases a) and b) in question 5).
Use separation of variables to solve to the
ODE: dx/dt= α_1 x-α_2 x^2;t>0;x(0)= x_0>0
You may use integral tables.
Find the equilibrium points of the ODE.
Evaluate the solution as t→∞
What conclusions can you make relative to the answers to questions 2 and 3.

© BrainMass Inc. brainmass.com March 22, 2019, 1:55 am ad1c9bdddf
https://brainmass.com/math/numerical-analysis/differential-equation-questions-555925

Solution Preview

1.
The equation is:
(1.1)
If we set:
(1.2)
Then:
(1.3)
The equation becomes:

(1.4)
This is a separable equation:


Continued:

(1.5)
And since we get after applying initial condition:

(1.6)
Thus:
(1.7)

And:
(1.8)

Note that:

(1.9)
So indeed, solution (1.8) is indeed the solution of the differential equation.

As we have (note that )
(1.10)
Therefore:
(1.11)
As N approaches the steady state solution

The equation is:
(1.12)
The steady state solution occurs when
(1.13)
Since is undefined at N=0 (though as ), the only other value for the steady state is:

(1.14)
Which is ...

Solution Summary

The expert examines ten differential equation questions.

$2.19