# 10 Differential Equation Questions

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.

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.