N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP.


Example 1

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 1)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 4 \, t )y'= {\left(t^{2} + 6 \, t + 5\right)} y - 2 \, t\hspace{2em}y( 7 )= -4\]

Answer.

\[( 4 ,\infty)\]


Example 2

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 2)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 5 \, t + 4 )y'= {\left(t^{2} + 5 \, t + 4\right)} y + 2 \, t\hspace{2em}y( -2 )= 5\]

Answer.

\[(-\infty, 1 )\]


Example 3

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 3)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - t - 6 )y'= -2 \, t^{3} + {\left(t^{2} - 3 \, t\right)} y\hspace{2em}y( 7 )= 2\]

Answer.

\[( 3 ,\infty)\]


Example 4

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 4)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - t - 2 )y'= {\left(t^{2} + 3 \, t - 4\right)} y - 2 \, t\hspace{2em}y( 3 )= 1\]

Answer.

\[( 2 ,\infty)\]


Example 5

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 5)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 6 \, t + 5 )y'= {\left(t^{2} + 5 \, t + 6\right)} y - t\hspace{2em}y( -4 )= -1\]

Answer.

\[( -5 , -1 )\]


Example 6

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 6)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 7 \, t + 6 )y'= {\left(t^{2} + 4 \, t + 4\right)} y + 3 \, t\hspace{2em}y( 8 )= 2\]

Answer.

\[( 6 ,\infty)\]


Example 7

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 7)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 5 \, t )y'= {\left(t^{2} - 5 \, t\right)} y\hspace{2em}y( 8 )= 3\]

Answer.

\[( 5 ,\infty)\]


Example 8

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 8)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 7 \, t + 6 )y'= {\left(t^{2} + 3 \, t - 4\right)} y + t\hspace{2em}y( 10 )= 1\]

Answer.

\[( 6 ,\infty)\]


Example 9

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 9)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - t - 2 )y'= 3 \, t^{3} + {\left(t^{2} + t - 2\right)} y\hspace{2em}y( -3 )= 0\]

Answer.

\[(-\infty, -1 )\]


Example 10

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 10)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 5 \, t + 4 )y'= {\left(t^{2} + 6 \, t + 5\right)} y\hspace{2em}y( 5 )= -3\]

Answer.

\[( 4 ,\infty)\]


Example 11

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 11)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + t - 2 )y'= -t^{2} + {\left(t^{2} + 8 \, t + 16\right)} y\hspace{2em}y( -3 )= -2\]

Answer.

\[(-\infty, -2 )\]


Example 12

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 12)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 4 )y'= 3 \, t^{3} + {\left(t^{2} - 4 \, t\right)} y\hspace{2em}y( 1 )= -2\]

Answer.

\[( -2 , 2 )\]


Example 13

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 13)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 3 \, t - 4 )y'= {\left(t^{2} + 2 \, t - 8\right)} y\hspace{2em}y( -6 )= -5\]

Answer.

\[(-\infty, -4 )\]


Example 14

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 14)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 8 \, t + 15 )y'= 2 \, t^{3} + {\left(t^{2} - t - 12\right)} y\hspace{2em}y( -4 )= -2\]

Answer.

\[( -5 , -3 )\]


Example 15

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 15)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 2 \, t - 3 )y'= -2 \, t^{2} + {\left(t^{2} + t - 6\right)} y\hspace{2em}y( 0 )= 2\]

Answer.

\[( -1 , 3 )\]


Example 16

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 16)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 2 \, t - 3 )y'= -3 \, t^{2} + {\left(t^{2} + 2 \, t - 15\right)} y\hspace{2em}y( -7 )= -5\]

Answer.

\[(-\infty, -3 )\]


Example 17

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 17)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + t - 2 )y'= {\left(t^{2} - 4 \, t\right)} y + 3 \, t\hspace{2em}y( 0 )= -3\]

Answer.

\[( -2 , 1 )\]


Example 18

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 18)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 5 \, t + 4 )y'= {\left(t^{2} + t - 2\right)} y - 2 \, t\hspace{2em}y( 6 )= 2\]

Answer.

\[( 4 ,\infty)\]


Example 19

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 19)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + t - 6 )y'= -t^{2} + {\left(t^{2} - 5 \, t\right)} y\hspace{2em}y( -5 )= 1\]

Answer.

\[(-\infty, -3 )\]


Example 20

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 20)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 2 \, t - 3 )y'= 3 \, t^{2} + {\left(t^{2} - 4 \, t + 3\right)} y\hspace{2em}y( 2 )= -3\]

Answer.

\[( 1 ,\infty)\]


Example 21

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 21)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 6 \, t + 8 )y'= {\left(t^{2} + 2 \, t - 3\right)} y\hspace{2em}y( 0 )= 2\]

Answer.

\[( -2 ,\infty)\]


Example 22

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 22)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 5 \, t + 4 )y'= t^{2} + {\left(t^{2} + 8 \, t + 16\right)} y\hspace{2em}y( 3 )= 0\]

Answer.

\[( -1 ,\infty)\]


Example 23

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 23)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 8 \, t + 15 )y'= -3 \, t^{2} + {\left(t^{2} + 6 \, t + 8\right)} y\hspace{2em}y( -2 )= 1\]

Answer.

\[( -3 ,\infty)\]


Example 24

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 24)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + t - 6 )y'= {\left(t^{2} + t - 2\right)} y\hspace{2em}y( 4 )= -3\]

Answer.

\[( 2 ,\infty)\]


Example 25

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 25)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - t - 6 )y'= {\left(t^{2} - 8 \, t + 15\right)} y - 2 \, t\hspace{2em}y( 2 )= -3\]

Answer.

\[( -2 , 3 )\]


Example 26

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 26)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 6 \, t + 5 )y'= 3 \, t^{2} + {\left(t^{2} + 2 \, t - 15\right)} y\hspace{2em}y( 4 )= 4\]

Answer.

\[( 1 , 5 )\]


Example 27

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 27)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 3 \, t - 4 )y'= -2 \, t^{3} + {\left(t^{2} - 3 \, t - 10\right)} y\hspace{2em}y( -7 )= -5\]

Answer.

\[(-\infty, -4 )\]


Example 28

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 28)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - t - 6 )y'= t^{2} + {\left(t^{2} - t - 6\right)} y\hspace{2em}y( -6 )= 3\]

Answer.

\[(-\infty, -2 )\]


Example 29

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 29)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 4 \, t )y'= {\left(t^{2} + 7 \, t + 10\right)} y + t\hspace{2em}y( 2 )= -3\]

Answer.

\[( 0 , 4 )\]


Example 30

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 30)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + t - 6 )y'= -3 \, t^{2} + {\left(t^{2} + 2 \, t - 15\right)} y\hspace{2em}y( 3 )= 3\]

Answer.

\[( 2 ,\infty)\]


Example 31

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 31)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 6 \, t + 5 )y'= -t^{2} + {\left(t^{2} + 4 \, t - 5\right)} y\hspace{2em}y( 1 )= 0\]

Answer.

\[( -1 ,\infty)\]


Example 32

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 32)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 4 )y'= {\left(t^{2} - 6 \, t + 9\right)} y + 2 \, t\hspace{2em}y( -6 )= 1\]

Answer.

\[(-\infty, -2 )\]


Example 33

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 33)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 7 \, t + 10 )y'= 2 \, t^{3} + {\left(t^{2} - 6 \, t + 9\right)} y\hspace{2em}y( -8 )= 1\]

Answer.

\[(-\infty, -5 )\]


Example 34

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 34)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 3 \, t - 4 )y'= 2 \, t^{2} + {\left(t^{2} + 3 \, t + 2\right)} y\hspace{2em}y( 8 )= -1\]

Answer.

\[( 4 ,\infty)\]


Example 35

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 35)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 3 \, t - 4 )y'= {\left(t^{2} - 5 \, t\right)} y\hspace{2em}y( 5 )= 3\]

Answer.

\[( 1 ,\infty)\]


Example 36

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 36)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 3 \, t )y'= {\left(t^{2} + 3 \, t\right)} y + 3 \, t\hspace{2em}y( 1 )= -1\]

Answer.

\[( 0 , 3 )\]


Example 37

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 37)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 4 \, t )y'= {\left(t^{2} - 3 \, t - 10\right)} y + t\hspace{2em}y( 2 )= -5\]

Answer.

\[( 0 , 4 )\]


Example 38

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 38)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 6 \, t + 8 )y'= {\left(t^{2} + 2 \, t - 15\right)} y - 3 \, t\hspace{2em}y( -8 )= -5\]

Answer.

\[(-\infty, -4 )\]


Example 39

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 39)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 2 \, t )y'= {\left(t^{2} - t - 2\right)} y - t\hspace{2em}y( -1 )= 5\]

Answer.

\[(-\infty, 0 )\]


Example 40

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 40)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 3 \, t )y'= -2 \, t^{3} + {\left(t^{2} + 8 \, t + 16\right)} y\hspace{2em}y( -1 )= -5\]

Answer.

\[( -3 , 0 )\]


Example 41

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 41)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + t - 6 )y'= 3 \, t^{2} + {\left(t^{2} - t - 12\right)} y\hspace{2em}y( 1 )= 2\]

Answer.

\[( -3 , 2 )\]


Example 42

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 42)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + t - 6 )y'= -2 \, t^{2} + {\left(t^{2} - 6 \, t + 9\right)} y\hspace{2em}y( -1 )= -5\]

Answer.

\[( -3 , 2 )\]


Example 43

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 43)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 4 \, t )y'= {\left(t^{2} - t - 2\right)} y + 3 \, t\hspace{2em}y( -1 )= 5\]

Answer.

\[( -4 , 0 )\]


Example 44

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 44)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 8 \, t + 15 )y'= {\left(t^{2} + 10 \, t + 25\right)} y - 3 \, t\hspace{2em}y( -4 )= 3\]

Answer.

\[( -5 , -3 )\]


Example 45

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 45)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 4 \, t )y'= 2 \, t^{2} + {\left(t^{2} + 3 \, t - 10\right)} y\hspace{2em}y( -2 )= -3\]

Answer.

\[(-\infty, 0 )\]


Example 46

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 46)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + t - 6 )y'= 3 \, t^{2} + {\left(t^{2} + 3 \, t + 2\right)} y\hspace{2em}y( 3 )= -4\]

Answer.

\[( 2 ,\infty)\]


Example 47

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 47)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} - 3 \, t )y'= 2 \, t^{2} + {\left(t^{2} + 7 \, t + 10\right)} y\hspace{2em}y( -2 )= -4\]

Answer.

\[(-\infty, 0 )\]


Example 48

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 48)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 4 \, t )y'= {\left(t^{2} - 6 \, t + 8\right)} y - 2 \, t\hspace{2em}y( 3 )= -5\]

Answer.

\[( 0 ,\infty)\]


Example 49

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 49)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 2 \, t )y'= -2 \, t^{3} + {\left(t^{2} + 4 \, t + 3\right)} y\hspace{2em}y( -6 )= 3\]

Answer.

\[(-\infty, -2 )\]


Example 50

N1 - Existence and uniqueness. Apply an existence and uniqueness theorem to an IVP. (ver. 50)

Find the largest interval for which the IVP Existence and Uniqueness Theorem guarantees a unique solution for the following IVP.

\[( t^{2} + 2 \, t )y'= t^{3} + {\left(t^{2} - 5 \, t + 4\right)} y\hspace{2em}y( -1 )= -2\]

Answer.

\[( -2 , 0 )\]