# Differential equations Problem Set 6

Please answer all of the following questions. The problems marked with an
asterisk will be

graded while the remained problems will be checked for completeness. Staple your
work to

this sheet of paper and indicate your answers clearly. Don’t forget your name
and please

circle your section number.

1. Section 10.1 (pp. 575–576) 5, 6, 7, 9, 15*, 16*.

2. Section 10.2 (pp. 585–587) 14*, 15, 17.

3. A second- order Cauchy -Euler equation has the form

where α are β are
constants (see Section 5.5). Homogeneous solutions of this equation may

be found in a similar manner as those for constant coefficient equations . The
purpose of

this exercise is to illustrate this point.

(a) Show that y = x^{r} is a solution of the Cauchy-Euler equation if r
satisfies the characteristic

polynomial

Let us now assume that r_{1} and r_{2 }are roots of the characteristic polynomial
obtained using

the quadratic formula . There are three cases to consider.

1. If r_{1} and r_{2} are real and
unequal, then the general solution has the form

2. If r_{1} and r_{2} are complex
conjugates, equal to λ±iμ say, then the general solution has

the form

3. If r_{1} = r_{2} = r, then the general
solution has the form

(b) Find general solutions for the following equations .

Parts (a) and (b) are “starred” problems.

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