Step by step solution to "Find r(t) if r′(t)=ti+etj+tek and r(0)=i+j+k"

Name: Find r(t) if r′(t)=ti+etj+tek and r(0)=i+j+k
Uploaded: 2022-02-23T17:22:57+00:00
Description: Find r(t) if r′(t)=ti+etj+tek and r(0)=i+j+k

illusiia

Answered question

2021-10-17

Find r(t) if

r^{'} (t) = t i + e^{t} j + t e^{k}

and

r (0) = i + j + k

Answer & Explanation

AGRFTr

Skilled2021-10-18Added 95 answers

r^{'} (t) = t i + e^{t} j + t e^{t} k

r (0) = i + j + k

Integrate each component of

r^{'} (t)

separetly. For the first component:

\int t d t = \frac{1}{2} t^{2} + c_{1}

The 1st component of r(0) is 1, so we can solve for

c_{1}

\frac{1}{2} {(0)}^{2} + c_{1} = 1

c_{1} = 1

So we have the first component for r(t):

(\frac{1}{2} t^{2} + 1) i

For the 2nd component:

\int e^{t} d t = e^{t} + c_{2}

The 2nd component of r(0) is 1, so we can solve for

c_{2}

e^{0} + c_{2} = 1

1 + c_{2} = 1

c_{2} = 0

So the 2nd component for r(t) is

e^{t} j

For the 3rd component use integration by parts:

u = t d u = d t

d v = e^{t} d t v = e^{t}

\int t e^{t} d t = u v - \int v d u

= t e^{t} - \int e^{t} d t

= t e^{t} - e^{t} + c_{3}

The 3rd component of

r (0)

is 1, so we can solve for

c_{3}

0 e^{0} - e^{0} + c_{3} = 1

- 1 + c_{3} = 1

c_{3} = 2

So the 2nd component for r(t) is:

(t e^{t} - e^{t} + 2) k

Result:

(\frac{1}{2} t^{2} + 1) i + e^{t} j + (t e^{t} - e^{t} + 2) k

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