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:
\[y_p( 3 )\approx 2.7 \]
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:
\[y_p( -3 )\approx 0.00 \]
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:
\[y_p( 0 )\approx -2.0 \]
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:
\[y_p( -4 )\approx 1.0 \]
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:
\[y_p( -3 )\approx 2.0 \]
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:
\[y_p( -3 )\approx 2.0 \]
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:
\[y_p( -3 )\approx 1.0 \]
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:
\[y_p( 3 )\approx 1.4 \]
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:
\[y_p( -2 )\approx -1.0 \]
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:
\[y_p( 4 )\approx -4.8 \]
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:
\[y_p( 0 )\approx 0.61 \]
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:
\[y_p( 4 )\approx 0.041 \]
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:
\[y_p( 0 )\approx 0.00 \]
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:
\[y_p( 0 )\approx -1.6 \]
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:
\[y_p( -1 )\approx 1.0 \]
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:
\[y_p( 0 )\approx 2.4 \]
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:
\[y_p( 0 )\approx 1.0 \]
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:
\[y_p( 3 )\approx -3.2 \]
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:
\[y_p( -1 )\approx 0.00 \]
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:
\[y_p( -1 )\approx 1.0 \]
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:
\[y_p( -2 )\approx -2.0 \]
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:
\[y_p( 1 )\approx -2.0 \]
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:
\[y_p( 4 )\approx 4.8 \]
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:
\[y_p( -4 )\approx -1.0 \]
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:
\[y_p( -3 )\approx -2.0 \]
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:
\[y_p( 3 )\approx -2.1 \]
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:
\[y_p( 0 )\approx -1.0 \]
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:
\[y_p( -1 )\approx -2.0 \]
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:
\[y_p( 3 )\approx 0.018 \]
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:
\[y_p( 4 )\approx -3.3 \]
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:
\[y_p( 1 )\approx 1.4 \]
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:
\[y_p( 0 )\approx 2.4 \]
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:
\[y_p( 1 )\approx -1.1 \]
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:
\[y_p( 1 )\approx -2.0 \]
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:
\[y_p( 4 )\approx -1.1 \]
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:
\[y_p( 2 )\approx 1.7 \]
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:
\[y_p( 1 )\approx 3.0 \times 10^{-9} \]
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:
\[y_p( 0 )\approx 0.00 \]
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:
\[y_p( 3 )\approx 1.7 \]
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:
\[y_p( 4 )\approx -1.1 \]
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:
\[y_p( -1 )\approx 1.0 \]
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:
\[y_p( 4 )\approx -2.8 \]
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:
\[y_p( 4 )\approx -1.0 \]
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:
\[y_p( 3 )\approx -3.9 \]
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:
\[y_p( -4 )\approx 2.0 \]
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:
\[y_p( -1 )\approx 0.00 \]
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:
\[y_p( -3 )\approx 2.0 \]
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:
\[y_p( -4 )\approx 0.00 \]
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:
\[y_p( -4 )\approx -1.0 \]
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:
\[y_p( -3 )\approx -2.0 \]