Download - ODE a Assignment7
-
8/12/2019 ODE a Assignment7
1/3
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?
-
8/12/2019 ODE a Assignment7
2/3
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 ====+ )(,)(,)(,
-
8/12/2019 ODE a Assignment7
3/3
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)