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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 Preview

The equation is:
If we set:
The equation becomes:

This is a separable equation:


And since we get after applying initial condition:



Note that:

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

As we have (note that )
As N approaches the steady state solution

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

Which is ...

Solution Summary

The expert examines ten differential equation questions.