## X2 - Existence/uniqueness theorem for linear IVPs

#### Example 1 π

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

$0 = {\left(t - 5\right)} {y''} e^{t} + {\left(t + 6\right)} t + {\left(t^{2} + 4\right)} y - 3 \, {y'} \hspace{2em} x( 7 )= 4$

$(5,+\infty)$

#### Example 2 π

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

$-{\left(t + 5\right)} {\left(t - 5\right)} - {\left(t + 1\right)} {y'''} = y e^{\left(-5 \, t\right)} + 2 \, {y''} \hspace{2em} x( -2 )= 2$

$(-\infty,-1)$

#### Example 3 π

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

$-{\left(t + 5\right)} {\left(t - 5\right)} {y''} = {\left(t - 1\right)} y e^{t} + t^{2} + 5 \, {y'} + 4 \hspace{2em} x( 3 )= -7$

$(-5,5)$

#### Example 4 π

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

$-{\left(t + 5\right)} {\left(t - 6\right)} {y'''} - y e^{\left(-t\right)} = t + 5 \, {y''} + 1 \hspace{2em} x( -1 )= -8$

$(-5,6)$

#### Example 5 π

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

$-4 \, {y''} = -{\left(t + 4\right)} {\left(t - 6\right)} {y'''} - y e^{\left(-t\right)} - {\left(t - 2\right)} e^{t} \hspace{2em} x( 3 )= 9$

$(-4,6)$

#### Example 6 π

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

$-{\left(t + 4\right)} {\left(t - 5\right)} {y''} - e^{\left(-5 \, t\right)} = t y - 3 \, {y'} \hspace{2em} x( 4 )= -7$

$(-4,5)$

#### Example 7 π

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

$y e^{t} + 2 \, {y''} = -{\left(t + 4\right)} {\left(t - 2\right)} - {\left(t - 6\right)} {y'''} \hspace{2em} x( 2 )= 8$

$(-\infty,6)$

#### Example 8 π

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

$-2 \, {y''} = -{\left(t + 2\right)} {\left(t - 5\right)} - {\left(t + 4\right)} {y'''} - y e^{\left(4 \, t\right)} \hspace{2em} x( -3 )= -7$

$(-4,+\infty)$

#### Example 9 π

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

$t^{2} + 9 = -{\left(t + 6\right)} {\left(t - 2\right)} y - {\left(t - 5\right)} {y'''} + 4 \, {y''} \hspace{2em} x( 1 )= 7$

$(-\infty,5)$

#### Example 10 π

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

${\left(t + 4\right)} e^{t} + y e^{t} - 2 \, {y'} = -{\left(t - 1\right)} {\left(t - 5\right)} {y''} \hspace{2em} x( 3 )= 7$

$(1,5)$

#### Example 11 π

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

${\left(t - 1\right)} {\left(t - 6\right)} + y e^{\left(-3 \, t\right)} = -{\left(t + 5\right)} {y''} e^{t} + 2 \, {y'} \hspace{2em} x( -2 )= -7$

$(-5,+\infty)$

#### Example 12 π

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

${\left(t - 5\right)} {y'''} e^{t} = -{\left(t + 5\right)} {\left(t - 2\right)} - y e^{\left(-2 \, t\right)} - {y''} \hspace{2em} x( 3 )= 6$

$(-\infty,5)$

#### Example 13 π

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

$-y e^{t} = {\left(t - 5\right)} {y'''} e^{t} + {\left(t + 6\right)} {\left(t + 2\right)} - 3 \, {y''} \hspace{2em} x( 2 )= 8$

$(-\infty,5)$

#### Example 14 π

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

$-{\left(t + 6\right)} {y'''} e^{t} + 4 \, {y''} = {\left(t + 1\right)} {\left(t - 5\right)} + {\left(t^{2} + 9\right)} y \hspace{2em} x( -10 )= -5$

$(-\infty,-6)$

#### Example 15 π

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

$-3 \, {y'} = {\left(t + 6\right)} {y''} e^{t} + {\left(t - 1\right)} {\left(t - 6\right)} + {\left(t^{2} + 4\right)} y \hspace{2em} x( -4 )= -8$

$(-6,+\infty)$

#### Example 16 π

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

$-{\left(t + 1\right)} {\left(t - 5\right)} - y e^{\left(-5 \, t\right)} = {\left(t + 5\right)} {y'''} - 3 \, {y''} \hspace{2em} x( -4 )= -6$

$(-5,+\infty)$

#### Example 17 π

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

$-t + 4 \, {y'} + 6 = {\left(t + 4\right)} {\left(t + 1\right)} {y''} + {\left(t^{2} + 4\right)} y \hspace{2em} x( -3 )= 2$

$(-4,-1)$

#### Example 18 π

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

${\left(t^{2} + 4\right)} y + {\left(t + 2\right)} e^{t} - 4 \, {y''} = -{\left(t + 6\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( 3 )= 10$

$(-6,6)$

#### Example 19 π

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

$-{\left(t + 6\right)} {y''} e^{t} + 5 \, {y'} = {\left(t - 1\right)} {\left(t - 5\right)} + y e^{\left(-t\right)} \hspace{2em} x( -8 )= -3$

$(-\infty,-6)$

#### Example 20 π

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

$-2 \, {y'} = {\left(t + 1\right)} {\left(t - 5\right)} + {\left(t^{2} + 4\right)} y + {\left(t + 5\right)} {y''} \hspace{2em} x( -8 )= -1$

$(-\infty,-5)$

#### Example 21 π

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

${\left(t - 1\right)} {\left(t - 6\right)} = -{\left(t + 6\right)} {y''} e^{t} - y e^{\left(2 \, t\right)} + 3 \, {y'} \hspace{2em} x( -10 )= -2$

$(-\infty,-6)$

#### Example 22 π

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

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

$(-4,1)$

#### Example 23 π

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

$0 = -{\left(t + 2\right)} {y''} e^{t} - {\left(t + 5\right)} {\left(t - 5\right)} - y e^{\left(-t\right)} - 5 \, {y'} \hspace{2em} x( 2 )= -5$

$(-2,+\infty)$

#### Example 24 π

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

${\left(t - 2\right)} {\left(t - 6\right)} {y'''} + {\left(t + 4\right)} y e^{t} = -t^{2} - 3 \, {y''} - 16 \hspace{2em} x( 5 )= -1$

$(2,6)$

#### Example 25 π

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

${\left(t - 1\right)} {y''} + 4 \, {y'} + e^{\left(-t\right)} = -{\left(t + 4\right)} {\left(t - 5\right)} y \hspace{2em} x( 4 )= -1$

$(1,+\infty)$

#### Example 26 π

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

$t^{2} - 4 \, {y''} + 9 = -{\left(t + 6\right)} {\left(t - 2\right)} {y'''} - {\left(t - 6\right)} y \hspace{2em} x( -5 )= 3$

$(-6,2)$

#### Example 27 π

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

${\left(t - 5\right)} {y'''} e^{t} + y e^{\left(-5 \, t\right)} = -{\left(t + 5\right)} {\left(t + 2\right)} + 3 \, {y''} \hspace{2em} x( 6 )= 1$

$(5,+\infty)$

#### Example 28 π

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

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

$(-5,6)$

#### Example 29 π

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

${\left(t + 6\right)} {\left(t + 2\right)} {y''} + y e^{\left(4 \, t\right)} = -{\left(t - 5\right)} e^{t} + 3 \, {y'} \hspace{2em} x( -4 )= 0$

$(-6,-2)$

#### Example 30 π

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

$-{\left(t - 5\right)} {y''} e^{t} = {\left(t + 6\right)} {\left(t + 1\right)} + y e^{\left(4 \, t\right)} + {y'} \hspace{2em} x( 1 )= 7$

$(-\infty,5)$

#### Example 31 π

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

$-{\left(t + 5\right)} {\left(t - 5\right)} - {\left(t + 1\right)} {y'''} = y e^{\left(-5 \, t\right)} + 2 \, {y''} \hspace{2em} x( -2 )= 2$

$(-\infty,-1)$

#### Example 32 π

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

${\left(t - 6\right)} {y''} + y e^{\left(-4 \, t\right)} + 2 \, {y'} = -{\left(t + 6\right)} t \hspace{2em} x( 7 )= 2$

$(6,+\infty)$

#### Example 33 π

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

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

$(2,6)$

#### Example 34 π

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

$-{\left(t + 6\right)} {\left(t - 5\right)} {y'''} + {y''} = t y e^{t} + e^{\left(-5 \, t\right)} \hspace{2em} x( 3 )= -8$

$(-6,5)$

#### Example 35 π

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

$y e^{\left(-2 \, t\right)} + t - 3 \, {y'} = -{\left(t + 6\right)} {\left(t - 6\right)} {y''} \hspace{2em} x( -4 )= 8$

$(-6,6)$

#### Example 36 π

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

${\left(t + 4\right)} {\left(t - 5\right)} + {\left(t^{2} + 25\right)} y = -{\left(t - 1\right)} {y''} - 5 \, {y'} \hspace{2em} x( -3 )= 5$

$(-\infty,1)$

#### Example 37 π

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

${\left(t^{2} + 25\right)} y - 3 \, {y''} = -{\left(t + 5\right)} {\left(t - 6\right)} - {\left(t - 1\right)} {y'''} \hspace{2em} x( -1 )= 4$

$(-\infty,1)$

#### Example 38 π

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

${\left(t + 6\right)} {\left(t - 6\right)} y + {\left(t + 1\right)} {y''} e^{t} - 3 \, {y'} = -t^{2} - 25 \hspace{2em} x( 1 )= -3$

$(-1,+\infty)$

#### Example 39 π

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

${\left(t - 5\right)} {y''} e^{t} + {\left(t + 5\right)} {\left(t + 2\right)} + y e^{\left(2 \, t\right)} + 3 \, {y'} = 0 \hspace{2em} x( 4 )= 7$

$(-\infty,5)$

#### Example 40 π

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

${\left(t - 5\right)} {y'''} e^{t} + {\left(t + 5\right)} {\left(t + 1\right)} + {\left(t^{2} + 9\right)} y = {y''} \hspace{2em} x( 1 )= 9$

$(-\infty,5)$

#### Example 41 π

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

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

$(-\infty,0)$

#### Example 42 π

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

${\left(t + 6\right)} {\left(t - 6\right)} y = -t^{2} - {\left(t + 2\right)} {y''} + 2 \, {y'} - 25 \hspace{2em} x( -3 )= -1$

$(-\infty,-2)$

#### Example 43 π

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

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

$(0,+\infty)$

#### Example 44 π

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

$-{\left(t - 2\right)} {\left(t - 6\right)} {y'''} - 2 \, {y''} = {\left(t + 6\right)} y + e^{\left(-3 \, t\right)} \hspace{2em} x( 5 )= 8$

$(2,6)$

#### Example 45 π

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

${\left(t + 5\right)} {y'''} e^{t} + t^{2} + 4 = -{\left(t - 2\right)} {\left(t - 5\right)} y + 5 \, {y''} \hspace{2em} x( -4 )= -8$

$(-5,+\infty)$

#### Example 46 π

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

$-{\left(t + 4\right)} {\left(t - 5\right)} {y''} - {\left(t + 2\right)} y e^{t} - t^{2} - 3 \, {y'} - 9 = 0 \hspace{2em} x( -3 )= 6$

$(-4,5)$

#### Example 47 π

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

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

$(-4,2)$

#### Example 48 π

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

${\left(t + 4\right)} {\left(t - 2\right)} {y''} + t^{2} + {\left(t - 5\right)} y + 9 = {y'} \hspace{2em} x( 0 )= 4$

$(-4,2)$

#### Example 49 π

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

$-{y'} = -{\left(t + 5\right)} {\left(t + 2\right)} y - {\left(t - 6\right)} {y''} - e^{t} \hspace{2em} x( 7 )= 4$

$(6,+\infty)$
$-{\left(t + 5\right)} {\left(t - 6\right)} {y''} - y e^{\left(-4 \, t\right)} - t + 2 = 3 \, {y'} \hspace{2em} x( 1 )= -7$
$(-5,6)$