In the above graphs you have examples of solutions that never cross the X-axis (
ID: 3108722 • Letter: I
Question
In the above graphs you have examples of solutions that never cross the X-axis (problem 3) and that g cross the x-axis infinitely many times (problems 1, 2, 5, and 6). It is hard to tell exactly how many times the solution crosses in problem 4 just from the graph. Is it possible to have a solution that crosses the axis exactly once? Give an example if you can or explain why it is impossible Is it possible to have a solution that crosses the axis exactly twice? Give an example if you can or explain 8. why it is impossible.Explanation / Answer
1. It is possible to have one solution that crosses the Axis only once for the following equation
x'' + 2x' + x = 0 and x(0) = 1 and x'(0) = 1
The solution of this equation is y = (1-x) e-x.
It is not possible to for the solution to cross the axis exactly 2 times because for the solution of 2nd order ODE we have only these choices for complementry function :
1. No roots of the equation (of D2, D etc) - which gives the solution having cos and sin which crosses the axis infinitely.
2. 2 real and equal roots which is disscussed in part 1 of the problem such solution will cross the axis only once.
3. 2 real and unequal roots will give solution like - a egx+ b efx which is = 0 for at most one value of x.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.