10 Differential Equation Questions
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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.
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Solution Summary
The expert examines ten differential equation questions.
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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 ...
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