# 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.

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