X2 - Existence/uniqueness theorem for linear IVPs


Example 1 πŸ”—

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

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

Answer:

\[(5,+\infty)\]


Example 2 πŸ”—

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

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

Answer:

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


Example 3 πŸ”—

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

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

Answer:

\[(-5,5)\]


Example 4 πŸ”—

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

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

Answer:

\[(-5,6)\]


Example 5 πŸ”—

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

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

Answer:

\[(-4,6)\]


Example 6 πŸ”—

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

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

Answer:

\[(-4,5)\]


Example 7 πŸ”—

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

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

Answer:

\[(-\infty,6)\]


Example 8 πŸ”—

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

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

Answer:

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


Example 9 πŸ”—

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

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

Answer:

\[(-\infty,5)\]


Example 10 πŸ”—

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

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

Answer:

\[(1,5)\]


Example 11 πŸ”—

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

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

Answer:

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


Example 12 πŸ”—

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

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

Answer:

\[(-\infty,5)\]


Example 13 πŸ”—

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

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

Answer:

\[(-\infty,5)\]


Example 14 πŸ”—

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

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

Answer:

\[(-\infty,-6)\]


Example 15 πŸ”—

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

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

Answer:

\[(-6,+\infty)\]


Example 16 πŸ”—

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

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

Answer:

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


Example 17 πŸ”—

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

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

Answer:

\[(-4,-1)\]


Example 18 πŸ”—

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

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

Answer:

\[(-6,6)\]


Example 19 πŸ”—

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

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

Answer:

\[(-\infty,-6)\]


Example 20 πŸ”—

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

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

Answer:

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


Example 21 πŸ”—

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

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

Answer:

\[(-\infty,-6)\]


Example 22 πŸ”—

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

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

Answer:

\[(-4,1)\]


Example 23 πŸ”—

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

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

Answer:

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


Example 24 πŸ”—

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

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

Answer:

\[(2,6)\]


Example 25 πŸ”—

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

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

Answer:

\[(1,+\infty)\]


Example 26 πŸ”—

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

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

Answer:

\[(-6,2)\]


Example 27 πŸ”—

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

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

Answer:

\[(5,+\infty)\]


Example 28 πŸ”—

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

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

Answer:

\[(-5,6)\]


Example 29 πŸ”—

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

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

Answer:

\[(-6,-2)\]


Example 30 πŸ”—

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

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

Answer:

\[(-\infty,5)\]


Example 31 πŸ”—

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

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

Answer:

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


Example 32 πŸ”—

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

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

Answer:

\[(6,+\infty)\]


Example 33 πŸ”—

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

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

Answer:

\[(2,6)\]


Example 34 πŸ”—

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

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

Answer:

\[(-6,5)\]


Example 35 πŸ”—

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

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

Answer:

\[(-6,6)\]


Example 36 πŸ”—

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

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

Answer:

\[(-\infty,1)\]


Example 37 πŸ”—

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

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

Answer:

\[(-\infty,1)\]


Example 38 πŸ”—

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

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

Answer:

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


Example 39 πŸ”—

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

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

Answer:

\[(-\infty,5)\]


Example 40 πŸ”—

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

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

Answer:

\[(-\infty,5)\]


Example 41 πŸ”—

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

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

Answer:

\[(-\infty,0)\]


Example 42 πŸ”—

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

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

Answer:

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


Example 43 πŸ”—

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

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

Answer:

\[(0,+\infty)\]


Example 44 πŸ”—

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

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

Answer:

\[(2,6)\]


Example 45 πŸ”—

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

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

Answer:

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


Example 46 πŸ”—

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

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

Answer:

\[(-4,5)\]


Example 47 πŸ”—

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

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

Answer:

\[(-4,2)\]


Example 48 πŸ”—

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

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

Answer:

\[(-4,2)\]


Example 49 πŸ”—

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

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

Answer:

\[(6,+\infty)\]


Example 50 πŸ”—

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

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

Answer:

\[(-5,6)\]