Second Order Linear Equations

PROBLEMS

In each of  Problems 1  through8 find the general solution of the given differential equation .

In each of Problems 9 through 16 find the solution of the given initial value problem . Sketch
the graph of the solution and describe its behavior as t increases.

17. Find a differential equation whose general solution is 
18. Find a differential equation whose general solution is 
19. Find the solution of the initial value problem

Plot the solution for 0 ≤ r ≤ 2 and determine its minimum value.

20. Find the solution of the initial value problem

Then determine the maximum value of the solution and also find the point where the
solution is zero.

21. Solve the initial value problem . Then find α so that
the solution approaches zero as .

22. Solve the initial value problem. Then find β so that the
solution approaches zero as .

In each of Problems 23 and 24 determine the value  of α if any . for which al l solutions tend to
zero as : also determine the values of α. if any , for which all (nonzero) solutions become
unbounded as 


25. Consider the initial value problem

where β > 0.

(a) Solve the initial value problem.
(b) Plot the solution when β= 1. Find the coordinates f the minimum point of
the solution in this case.
(c) Find the smallest value of β for which the solution has no minimum point.