F1 - Direction fields for first-order ODEs


Example 1 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \cos\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 3 \).

\[ {y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 2 \]

Answer:

\[y_p( 3 )\approx 2.7 \]


Example 2 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left(t + {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -3 \).

\[ {y'} = \sin\left(t + {y}\right) \hspace{2em} y( -1 )= 0 \]

Answer:

\[y_p( -3 )\approx 0.00 \]


Example 3 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \cos\left({y} + t\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 0 \).

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

Answer:

\[y_p( 0 )\approx -2.0 \]


Example 4 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -4 \).

\[ {y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -2 )= 1 \]

Answer:

\[y_p( -4 )\approx 1.0 \]


Example 5 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left({y} + t\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -3 \).

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

Answer:

\[y_p( -3 )\approx 2.0 \]


Example 6 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \cos\left({y} + t\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -3 \).

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

Answer:

\[y_p( -3 )\approx 2.0 \]


Example 7 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -3 \).

\[ {y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( -1 )= 1 \]

Answer:

\[y_p( -3 )\approx 1.0 \]


Example 8 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 3 \).

\[ {y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 1 )= 2 \]

Answer:

\[y_p( 3 )\approx 1.4 \]


Example 9 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left({y} + t\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -2 \).

\[ {y'} = \sin\left({y} + t\right) \hspace{2em} y( 0 )= -1 \]

Answer:

\[y_p( -2 )\approx -1.0 \]


Example 10 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 4 \).

\[ {y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= -1 \]

Answer:

\[y_p( 4 )\approx -4.8 \]


Example 11 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 0 \).

\[ {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -2 )= 0 \]

Answer:

\[y_p( 0 )\approx 0.61 \]


Example 12 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left({y} + t\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 4 \).

\[ {y'} = \sin\left({y} + t\right) \hspace{2em} y( 2 )= 1 \]

Answer:

\[y_p( 4 )\approx 0.041 \]


Example 13 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 0 \).

\[ {y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= 0 \]

Answer:

\[y_p( 0 )\approx 0.00 \]


Example 14 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 0 \).

\[ {y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -2 )= -2 \]

Answer:

\[y_p( 0 )\approx -1.6 \]


Example 15 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -1 \).

\[ {y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 1 \]

Answer:

\[y_p( -1 )\approx 1.0 \]


Example 16 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 0 \).

\[ {y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \hspace{2em} y( -2 )= 2 \]

Answer:

\[y_p( 0 )\approx 2.4 \]


Example 17 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left({y} + t\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 0 \).

\[ {y'} = \sin\left({y} + t\right) \hspace{2em} y( 2 )= 1 \]

Answer:

\[y_p( 0 )\approx 1.0 \]


Example 18 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 3 \).

\[ {y'} = \frac{1}{9} \, t {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= -1 \]

Answer:

\[y_p( 3 )\approx -3.2 \]


Example 19 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -1 \).

\[ {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= 0 \]

Answer:

\[y_p( -1 )\approx 0.00 \]


Example 20 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left({y} + t\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -1 \).

\[ {y'} = \sin\left({y} + t\right) \hspace{2em} y( 1 )= 1 \]

Answer:

\[y_p( -1 )\approx 1.0 \]


Example 21 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -2 \).

\[ {y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 0 )= -2 \]

Answer:

\[y_p( -2 )\approx -2.0 \]


Example 22 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 1 \).

\[ {y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( -1 )= -2 \]

Answer:

\[y_p( 1 )\approx -2.0 \]


Example 23 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 4 \).

\[ {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= 1 \]

Answer:

\[y_p( 4 )\approx 4.8 \]


Example 24 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left(t + {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -4 \).

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

Answer:

\[y_p( -4 )\approx -1.0 \]


Example 25 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -3 \).

\[ {y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \hspace{2em} y( -1 )= -2 \]

Answer:

\[y_p( -3 )\approx -2.0 \]


Example 26 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 3 \).

\[ {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 1 )= -1 \]

Answer:

\[y_p( 3 )\approx -2.1 \]


Example 27 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \cos\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 0 \).

\[ {y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 2 )= -1 \]

Answer:

\[y_p( 0 )\approx -1.0 \]


Example 28 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \cos\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -1 \).

\[ {y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -2 \]

Answer:

\[y_p( -1 )\approx -2.0 \]


Example 29 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \cos\left({y} + t\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 3 \).

\[ {y'} = \cos\left({y} + t\right) \hspace{2em} y( 1 )= 2 \]

Answer:

\[y_p( 3 )\approx 0.018 \]


Example 30 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 4 \).

\[ {y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= -1 \]

Answer:

\[y_p( 4 )\approx -3.3 \]


Example 31 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(cos(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \cos\left({y} + t\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 1 \).

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

Answer:

\[y_p( 1 )\approx 1.4 \]


Example 32 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 0 \).

\[ {y'} = -\frac{1}{3} \, t - \frac{1}{15} \, {y} \hspace{2em} y( -2 )= 2 \]

Answer:

\[y_p( 0 )\approx 2.4 \]


Example 33 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 1 \).

\[ {y'} = \frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= -1 \]

Answer:

\[y_p( 1 )\approx -1.1 \]


Example 34 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 1 \).

\[ {y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( -1 )= -2 \]

Answer:

\[y_p( 1 )\approx -2.0 \]


Example 35 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 4 \).

\[ {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2 \]

Answer:

\[y_p( 4 )\approx -1.1 \]


Example 36 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \cos\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 2 \).

\[ {y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 0 )= 0 \]

Answer:

\[y_p( 2 )\approx 1.7 \]


Example 37 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 1 \).

\[ {y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( -1 )= 0 \]

Answer:

\[y_p( 1 )\approx 3.0 \times 10^{-9} \]


Example 38 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 0 \).

\[ {y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( 2 )= 0 \]

Answer:

\[y_p( 0 )\approx 0.00 \]


Example 39 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(cos(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \cos\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 3 \).

\[ {y'} = \cos\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 0 \]

Answer:

\[y_p( 3 )\approx 1.7 \]


Example 40 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 4 \).

\[ {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( 2 )= -2 \]

Answer:

\[y_p( 4 )\approx -1.1 \]


Example 41 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -1 \).

\[ {y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em} y( 1 )= 1 \]

Answer:

\[y_p( -1 )\approx 1.0 \]


Example 42 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 4 \).

\[ {y'} = \frac{1}{9} \, {y} t - \frac{1}{3} \, t \hspace{2em} y( 2 )= 0 \]

Answer:

\[y_p( 4 )\approx -2.8 \]


Example 43 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 4 \).

\[ {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( 2 )= 1 \]

Answer:

\[y_p( 4 )\approx -1.0 \]


Example 44 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= 3 \).

\[ {y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= -2 \]

Answer:

\[y_p( 3 )\approx -3.9 \]


Example 45 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(y/15-t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -4 \).

\[ {y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y} \hspace{2em} y( -2 )= 2 \]

Answer:

\[y_p( -4 )\approx 2.0 \]


Example 46 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left(\frac{1}{2} \, {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -1 \).

\[ {y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em} y( 1 )= 0 \]

Answer:

\[y_p( -1 )\approx 0.00 \]


Example 47 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t/3-y/15, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -3 \).

\[ {y'} = -\frac{1}{15} \, {y} - \frac{1}{3} \, t \hspace{2em} y( -1 )= 2 \]

Answer:

\[y_p( -3 )\approx 2.0 \]


Example 48 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -4 \).

\[ {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -2 )= 0 \]

Answer:

\[y_p( -4 )\approx 0.00 \]


Example 49 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(sin(t+y), (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \sin\left(t + {y}\right) \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -4 \).

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

Answer:

\[y_p( -4 )\approx -1.0 \]


Example 50 πŸ”—

Use https://sagecell.sagemath.org/ to run the SageMath code t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5)) producing the direction field for the ODE \( {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \).

Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -3 \).

\[ {y'} = \frac{1}{9} \, {y} t + \frac{1}{3} \, t \hspace{2em} y( -1 )= -2 \]

Answer:

\[y_p( -3 )\approx -2.0 \]