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(4) (Section 4.5, #2) Show that all solutions to the system of linear differential equations:dr3xdtdy-2ydtare linear combinations of the two solutions:and V (t)- 2tU(t) e3t

Question

It is linear algebra. Please see the attachment

(4) (Section 4.5, #2) Show that all solutions to the system of linear differential equations:
dr
3x
dt
dy
-2y
dt
are linear combinations of the two solutions:
and V (t)
- 2t
U(t) e3t
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(4) (Section 4.5, #2) Show that all solutions to the system of linear differential equations: dr 3x dt dy -2y dt are linear combinations of the two solutions: and V (t) - 2t U(t) e3t

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Step 1

Given system of first order linear differential equations is shown below

dx
= 3x
dt
dy2y
dt
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dx = 3x dt dy2y dt

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Step 2

To show that all solutions of given system of linear differential equations can be written as a linear combination of below two solution.

U (t)
and
0
V(t) e
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U (t) and 0 V(t) e

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Step 3

Solution of given differential eq...

dx
= 3x
dt
dx
dt
3х
1dx
3
x
1
In x =t +C
3
In x 3t3C
x =e3+3C
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dx = 3x dt dx dt 3х 1dx 3 x 1 In x =t +C 3 In x 3t3C x =e3+3C

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