ode a assignment7

8/12/2019 ODE a Assignment7

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ODE A7 - 1

ODE Chapter 7 AAssignment

7-1 Use the Wronskian to determine whether or not the following sets of functions arelinearly independent or linearly dependent

(a) }tt ete , (g) { })sin(),cos(),sin( ttt 2

(b) }22 223 tttt ,, (h) }tett ),ln(, (c) }12 2 ),(sin),cos( tt (i) })sin(),sin( tette tt

(d) }ttt eteet ,,2 (j)

+ 1

1

1

1

1

12

ttt,,

(e)

+ 1

1

1

1

tt, (k) })(sin),sin(),sin( ttt 33

(f) { }12 ),sin(),cos( tt

7-2 Decide whether or not the following sets S are fundamental sets of solutions to the

given differential equation

(a) } 093533 =++= yyyyteeeS ttt ,,, (b) } 02616955 =++= yyyyteteeS ttt ,)cos(),sin(, (c) { } 081472 =+++= yyyyeeS tt ,, (d) } 01503529115523 =++= yyyyyteeeeS ivtttt )(,,,, (e) { } 0632 == yyyeeS tt ,, (f) } 042 22 == yyeetS tt ,,),cosh(

7-3 Can }821 2 ttt ,,, be the fundamental set of solutions to a constant coefficientfourth order linear ODE?

7-4 Can })sin(),sin(),cos(,, tttee tt 2 be the fundamental set of solutions to a constantcoefficient fifth order linear ODE?

7-5 Can })sin(),cos(,, teteee tttt 42 be the fundamental set of solutions to a constantcoefficient fifth order linear ODE?

7-6 Can }ttttt etteeee 242 ,,,, be the fundamental set of solutions to a constantcoefficient fourth order linear ODE?


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ODE A7 - 2

7-7 Can })sin(),cos(,, teteee tttt 42 be the fundamental set of solutions to a constantcoefficient fourth order linear ODE?

7-8 Can })sin(),cos(,, teteee tttt 42 be the fundamental set of solutions to a constantcoefficient fourth order linear ODE?

7-9 In the following problems verify that the functions )(xy1 and )(xy2 are solutions of

the given differential equation. Determine the values ofxfor which 1y and 2y (and

3y ) are linearly independent.

(a) xxx eexyexyyyy 221023 ===+ )(,)(, ,

(b) xx exxyexyyyy 222

1 32044 )()(,)(, ===+

(c) )cosh()(,)(),sinh()(, xxyexyxxyyy x

3309 33

21 ====

(d) )cos()(),sin()(, xxyxxyyy 3309 21 ===+

(e) )ln()(,)(, xxxyxxyyyxyx2

2

2

1

2 043 ===+

(f) )cos()(),sin()(, xexyxexyyyy xx 330134 22

2

1 ===+

(g)xxx

exxyxexyexyyyyy22

3

2

2

2

108126 ====+ )(,)(,)(,


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ODE A7 - 3

Answers to ODE Chapter 7 AAssignment

7-1 (a) Independent (g) Independent

(b) Dependent (h) Independent

(c) Dependent (i) Independent

(d) Independent (j) Dependent(e) Independent (k) Dependent

(f) Independent

7-2 (a) A fundamental set of solutions.

(b) A fundamental set of solutions.

(c) Not a fundamental set of solutions because there are not enough functions.

(d) A fundamental set of solutions.

(e) Not a fundamental set of solutions ast

e2 is not a solution.

(f) Not a fundamental set of solutions as they are not linearly independent.

7-3 No, they are not linearly independent.

7-4 No, the )sin( t2 solution comes from a root of 2iof the characteristic equation. Since

the coefficients of the characteristic equation are real, this means that -2 iis also a root,

and then the characteristic equation must have 6 roots, which is impossible for a

degree 5 equation.

7-5 No, there are not enough functions for a fundamental set of solutions.

7-6 No, there are too many functions. The set indicates that the characteristic equation has

5 roots, which is impossible for a degree 4 equation.

7-7 Yes. The root of the characteristic equation would be .,, i1241

7-8 No, the functions )cos(te t and )sin(te

tpoint to two non-conjugate solutions to the

characteristic equation. Hence the characteristic equation would have to have 6

solutions which is impossible for a fourth order equation.

7-9 (a)

ode a assignment7

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