X3 - Existence/uniqueness theorem for first-order IVPs


Example 1 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 6 \, {\left(3 \, {y} + 4 \, t + 38\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -6 \]

Answer:

\(F(t,y)= 6 \, {\left(3 \, {y} + 4 \, t + 38\right)}^{\frac{1}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{6}{{\left(3 \, {y} + 4 \, t + 38\right)}^{\frac{2}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 2 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 3 \, {\left(2 \, {y} + 4 \, t - 24\right)}^{\frac{7}{5}} \hspace{2em} x( 3 )= 6 \]

Answer:

\(F(t,y)= 3 \, {\left(2 \, {y} + 4 \, t - 24\right)}^{\frac{7}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{42}{5} \, {\left(2 \, {y} + 4 \, t - 24\right)}^{\frac{2}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 3 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(6 \, t + 3 \, {y} - 36\right)}^{\frac{8}{3}} \hspace{2em} x( 3 )= 6 \]

Answer:

\(F(t,y)= 2 \, {\left(6 \, t + 3 \, {y} - 36\right)}^{\frac{8}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 16 \, {\left(6 \, t + 3 \, {y} - 36\right)}^{\frac{5}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 4 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 3 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{2}{{\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{2}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 5 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 6 \, {\left(2 \, {y} - 4 \, t + 10\right)}^{\frac{1}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{4}{{\left(2 \, {y} - 4 \, t + 10\right)}^{\frac{2}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 6 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 3 \, {\left(3 \, {y} + 6 \, t + 27\right)}^{\frac{1}{5}} \hspace{2em} x( -6 )= 3 \]

Answer:

\(F(t,y)= 3 \, {\left(3 \, {y} + 6 \, t + 27\right)}^{\frac{1}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{9}{5 \, {\left(3 \, {y} + 6 \, t + 27\right)}^{\frac{4}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 7 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 6 \, {\left(5 \, t + 2 \, {y} - 14\right)}^{\frac{8}{5}} \hspace{2em} x( 2 )= 2 \]

Answer:

\(F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} - 14\right)}^{\frac{8}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{96}{5} \, {\left(5 \, t + 2 \, {y} - 14\right)}^{\frac{3}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 8 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{7}{5}} \hspace{2em} x( 4 )= 5 \]

Answer:

\(F(t,y)= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{7}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{63}{5} \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 9 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 6 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{2}{3}} \hspace{2em} x( 6 )= -3 \]

Answer:

\(F(t,y)= 6 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{2}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{8}{{\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{1}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 10 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 6 \, {\left(2 \, {y} - 4 \, t + 4\right)}^{\frac{1}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{4}{{\left(2 \, {y} - 4 \, t + 4\right)}^{\frac{2}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 11 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 2 \, {\left(3 \, {y} - 2 \, t + 25\right)}^{\frac{1}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{6}{5 \, {\left(3 \, {y} - 2 \, t + 25\right)}^{\frac{4}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 12 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 4 \, {\left(3 \, {y} + 6 \, t\right)}^{\frac{1}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{12}{5 \, {\left(3 \, {y} + 6 \, t\right)}^{\frac{4}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 13 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 3 \, {\left(3 \, {y} - 6 \, t + 18\right)}^{\frac{1}{5}} \hspace{2em} x( 6 )= 6 \]

Answer:

\(F(t,y)= 3 \, {\left(3 \, {y} - 6 \, t + 18\right)}^{\frac{1}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{9}{5 \, {\left(3 \, {y} - 6 \, t + 18\right)}^{\frac{4}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 14 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(3 \, {y} - 6 \, t - 18\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= -4 \]

Answer:

\(F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t - 18\right)}^{\frac{8}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 16 \, {\left(3 \, {y} - 6 \, t - 18\right)}^{\frac{5}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 15 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{8}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{48}{5} \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{3}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 16 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 4 \, {\left(2 \, {y} + 3 \, t - 2\right)}^{\frac{4}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{32}{3} \, {\left(2 \, {y} + 3 \, t - 2\right)}^{\frac{1}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 17 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{2}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{8}{{\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{1}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 18 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 6 \, {\left(3 \, {y} - 4 \, t + 30\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -2 \]

Answer:

\(F(t,y)= 6 \, {\left(3 \, {y} - 4 \, t + 30\right)}^{\frac{4}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 24 \, {\left(3 \, {y} - 4 \, t + 30\right)}^{\frac{1}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 19 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(2 \, {y} - 3 \, t - 27\right)}^{\frac{1}{5}} \hspace{2em} x( -5 )= 6 \]

Answer:

\(F(t,y)= 2 \, {\left(2 \, {y} - 3 \, t - 27\right)}^{\frac{1}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{4}{5 \, {\left(2 \, {y} - 3 \, t - 27\right)}^{\frac{4}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 20 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 5 \, {\left(2 \, t + 2 \, {y} + 14\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= -4 \]

Answer:

\(F(t,y)= 5 \, {\left(2 \, t + 2 \, {y} + 14\right)}^{\frac{2}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{20}{3 \, {\left(2 \, t + 2 \, {y} + 14\right)}^{\frac{1}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 21 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 5 \, {\left(-6 \, t + 3 \, {y} + 42\right)}^{\frac{2}{5}} \hspace{2em} x( 5 )= -4 \]

Answer:

\(F(t,y)= 5 \, {\left(-6 \, t + 3 \, {y} + 42\right)}^{\frac{2}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{6}{{\left(-6 \, t + 3 \, {y} + 42\right)}^{\frac{3}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 22 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 5 \, {\left(3 \, {y} + 2 \, t + 14\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= -2 \]

Answer:

\(F(t,y)= 5 \, {\left(3 \, {y} + 2 \, t + 14\right)}^{\frac{1}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{3}{{\left(3 \, {y} + 2 \, t + 14\right)}^{\frac{4}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 23 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 2 \]

Answer:

\(F(t,y)= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{8}{3 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{1}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 24 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 5 \, {\left(2 \, {y} - 4 \, t + 2\right)}^{\frac{8}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 16 \, {\left(2 \, {y} - 4 \, t + 2\right)}^{\frac{3}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 25 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 4 \, {\left(-4 \, t + 3 \, {y} - 25\right)}^{\frac{4}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{48}{5 \, {\left(-4 \, t + 3 \, {y} - 25\right)}^{\frac{1}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 26 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(3 \, {y} + 3 \, t + 33\right)}^{\frac{2}{5}} \hspace{2em} x( -6 )= -5 \]

Answer:

\(F(t,y)= 2 \, {\left(3 \, {y} + 3 \, t + 33\right)}^{\frac{2}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{12}{5 \, {\left(3 \, {y} + 3 \, t + 33\right)}^{\frac{3}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 27 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 5 \, {\left(2 \, {y} + 5 \, t - 23\right)}^{\frac{7}{3}} \hspace{2em} x( 3 )= 4 \]

Answer:

\(F(t,y)= 5 \, {\left(2 \, {y} + 5 \, t - 23\right)}^{\frac{7}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{70}{3} \, {\left(2 \, {y} + 5 \, t - 23\right)}^{\frac{4}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 28 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 4 \, {\left(-2 \, t + 2 \, {y} - 2\right)}^{\frac{4}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{32}{5 \, {\left(-2 \, t + 2 \, {y} - 2\right)}^{\frac{1}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 29 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(3 \, {y} + 4 \, t + 18\right)}^{\frac{1}{3}} \hspace{2em} x( -6 )= 2 \]

Answer:

\(F(t,y)= 2 \, {\left(3 \, {y} + 4 \, t + 18\right)}^{\frac{1}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{2}{{\left(3 \, {y} + 4 \, t + 18\right)}^{\frac{2}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 30 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 2 \, {\left(-2 \, t + 3 \, {y}\right)}^{\frac{7}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 14 \, {\left(-2 \, t + 3 \, {y}\right)}^{\frac{4}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 31 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 6 \, {\left(2 \, {y} + 4 \, t + 10\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= 3 \]

Answer:

\(F(t,y)= 6 \, {\left(2 \, {y} + 4 \, t + 10\right)}^{\frac{8}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 32 \, {\left(2 \, {y} + 4 \, t + 10\right)}^{\frac{5}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 32 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(2 \, {y} - 4 \, t + 20\right)}^{\frac{7}{3}} \hspace{2em} x( 3 )= -4 \]

Answer:

\(F(t,y)= 2 \, {\left(2 \, {y} - 4 \, t + 20\right)}^{\frac{7}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{28}{3} \, {\left(2 \, {y} - 4 \, t + 20\right)}^{\frac{4}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 33 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 5 \, {\left(3 \, {y} + 3 \, t + 27\right)}^{\frac{8}{5}} \hspace{2em} x( -6 )= -3 \]

Answer:

\(F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t + 27\right)}^{\frac{8}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 24 \, {\left(3 \, {y} + 3 \, t + 27\right)}^{\frac{3}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 34 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 6 \, {\left(2 \, {y} + 3 \, t - 19\right)}^{\frac{1}{5}} \hspace{2em} x( 3 )= 5 \]

Answer:

\(F(t,y)= 6 \, {\left(2 \, {y} + 3 \, t - 19\right)}^{\frac{1}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{12}{5 \, {\left(2 \, {y} + 3 \, t - 19\right)}^{\frac{4}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 35 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t + 8\right)}^{\frac{2}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{16}{3 \, {\left(2 \, {y} - 2 \, t + 8\right)}^{\frac{1}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 36 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 3 \, {\left(3 \, {y} + 4 \, t - 42\right)}^{\frac{8}{3}} \hspace{2em} x( 6 )= 6 \]

Answer:

\(F(t,y)= 3 \, {\left(3 \, {y} + 4 \, t - 42\right)}^{\frac{8}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 24 \, {\left(3 \, {y} + 4 \, t - 42\right)}^{\frac{5}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 37 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 4 \, {\left(3 \, {y} - 4 \, t + 26\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -2 \]

Answer:

\(F(t,y)= 4 \, {\left(3 \, {y} - 4 \, t + 26\right)}^{\frac{2}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{8}{{\left(3 \, {y} - 4 \, t + 26\right)}^{\frac{1}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 38 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 5 \, {\left(3 \, {y} + 5 \, t + 48\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -6 \]

Answer:

\(F(t,y)= 5 \, {\left(3 \, {y} + 5 \, t + 48\right)}^{\frac{8}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 40 \, {\left(3 \, {y} + 5 \, t + 48\right)}^{\frac{5}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 39 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 6 \, {\left(-3 \, t + 2 \, {y} + 22\right)}^{\frac{1}{3}} \hspace{2em} x( 6 )= -2 \]

Answer:

\(F(t,y)= 6 \, {\left(-3 \, t + 2 \, {y} + 22\right)}^{\frac{1}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{4}{{\left(-3 \, t + 2 \, {y} + 22\right)}^{\frac{2}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 40 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(3 \, {y} - 6 \, t + 30\right)}^{\frac{7}{5}} \hspace{2em} x( 2 )= -6 \]

Answer:

\(F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t + 30\right)}^{\frac{7}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{42}{5} \, {\left(3 \, {y} - 6 \, t + 30\right)}^{\frac{2}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 41 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 4 \, {\left(2 \, t + 3 \, {y} - 7\right)}^{\frac{1}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{4}{{\left(2 \, t + 3 \, {y} - 7\right)}^{\frac{2}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 42 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 5 \, {\left(3 \, {y} + 3 \, t - 18\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= 2 \]

Answer:

\(F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 18\right)}^{\frac{7}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= 35 \, {\left(3 \, {y} + 3 \, t - 18\right)}^{\frac{4}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 43 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 4 \, {\left(3 \, {y} - 5 \, t - 42\right)}^{\frac{8}{5}} \hspace{2em} x( -6 )= 4 \]

Answer:

\(F(t,y)= 4 \, {\left(3 \, {y} - 5 \, t - 42\right)}^{\frac{8}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{96}{5} \, {\left(3 \, {y} - 5 \, t - 42\right)}^{\frac{3}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 44 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 4 \, {\left(2 \, {y} + 6 \, t + 8\right)}^{\frac{4}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{32}{3} \, {\left(2 \, {y} + 6 \, t + 8\right)}^{\frac{1}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 45 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{7}{5}} \hspace{2em} x( -3 )= -4 \]

Answer:

\(F(t,y)= 2 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{7}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{42}{5} \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{2}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 46 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 3 \, {\left(3 \, {y} - 2 \, t - 13\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= 3 \]

Answer:

\(F(t,y)= 3 \, {\left(3 \, {y} - 2 \, t - 13\right)}^{\frac{4}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{36}{5 \, {\left(3 \, {y} - 2 \, t - 13\right)}^{\frac{1}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 47 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 5 \, {\left(3 \, {y} + 2 \, t + 1\right)}^{\frac{2}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{10}{{\left(3 \, {y} + 2 \, t + 1\right)}^{\frac{1}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 48 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

\[ y'= 2 \, {\left(2 \, {y} + 5 \, t + 14\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= 3 \]

Answer:

\(F(t,y)= 2 \, {\left(2 \, {y} + 5 \, t + 14\right)}^{\frac{8}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{32}{3} \, {\left(2 \, {y} + 5 \, t + 14\right)}^{\frac{5}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.


Example 49 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 5 \, {\left(2 \, {y} - 6 \, t - 16\right)}^{\frac{1}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{2}{{\left(2 \, {y} - 6 \, t - 16\right)}^{\frac{4}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.


Example 50 πŸ”—

Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.

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

Answer:

\(F(t,y)= 2 \, {\left(-5 \, t + 2 \, {y} + 16\right)}^{\frac{8}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.

\(F_y= \frac{32}{5} \, {\left(-5 \, t + 2 \, {y} + 16\right)}^{\frac{3}{5}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.