## F1 - Direction fields for first-order ODEs

#### Example 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 $$t= 3$$.

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

$y_p( 3 )\approx 2.7$

#### Example 2 🔗

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 $$t= -3$$.

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

$y_p( -3 )\approx 0.00$

#### Example 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 $$t= 0$$.

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

$y_p( 0 )\approx -2.0$

#### Example 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 $${y'} = -\frac{1}{3} \, t + \frac{1}{15} \, {y}$$.

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 $$t= -4$$.

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

$y_p( -4 )\approx 1.0$

#### Example 5 🔗

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 $$t= -3$$.

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

$y_p( -3 )\approx 2.0$

#### Example 6 🔗

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 $$t= -3$$.

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

$y_p( -3 )\approx 2.0$

#### Example 7 🔗

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 $$t= -3$$.

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

$y_p( -3 )\approx 1.0$

#### Example 8 🔗

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 $$t= 3$$.

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

$y_p( 3 )\approx 1.4$

#### Example 9 🔗

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 $$t= -2$$.

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

$y_p( -2 )\approx -1.0$

#### Example 10 🔗

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 $$t= 4$$.

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

$y_p( 4 )\approx -4.8$

#### Example 11 🔗

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 $$t= 0$$.

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

$y_p( 0 )\approx 0.61$

#### Example 12 🔗

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 $$t= 4$$.

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

$y_p( 4 )\approx 0.041$

#### Example 13 🔗

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 $$t= 0$$.

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

$y_p( 0 )\approx 0.00$

#### Example 14 🔗

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 $$t= 0$$.

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

$y_p( 0 )\approx -1.6$

#### Example 15 🔗

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 $$t= -1$$.

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

$y_p( -1 )\approx 1.0$

#### Example 16 🔗

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 $$t= 0$$.

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

$y_p( 0 )\approx 2.4$

#### Example 17 🔗

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 $$t= 0$$.

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

$y_p( 0 )\approx 1.0$

#### Example 18 🔗

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 $$t= 3$$.

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

$y_p( 3 )\approx -3.2$

#### Example 19 🔗

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 $$t= -1$$.

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

$y_p( -1 )\approx 0.00$

#### Example 20 🔗

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 $$t= -1$$.

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

$y_p( -1 )\approx 1.0$

#### Example 21 🔗

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 $$t= -2$$.

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

$y_p( -2 )\approx -2.0$

#### Example 22 🔗

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 $$t= 1$$.

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

$y_p( 1 )\approx -2.0$

#### Example 23 🔗

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 $$t= 4$$.

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

$y_p( 4 )\approx 4.8$

#### Example 24 🔗

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 $$t= -4$$.

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

$y_p( -4 )\approx -1.0$

#### Example 25 🔗

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 $$t= -3$$.

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

$y_p( -3 )\approx -2.0$

#### Example 26 🔗

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 $$t= 3$$.

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

$y_p( 3 )\approx -2.1$

#### Example 27 🔗

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 $$t= 0$$.

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

$y_p( 0 )\approx -1.0$

#### Example 28 🔗

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 $$t= -1$$.

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

$y_p( -1 )\approx -2.0$

#### Example 29 🔗

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 $$t= 3$$.

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

$y_p( 3 )\approx 0.018$

#### Example 30 🔗

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 $$t= 4$$.

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

$y_p( 4 )\approx -3.3$

#### Example 31 🔗

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 $$t= 1$$.

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

$y_p( 1 )\approx 1.4$

#### Example 32 🔗

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 $$t= 0$$.

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

$y_p( 0 )\approx 2.4$

#### Example 33 🔗

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 $$t= 1$$.

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

$y_p( 1 )\approx -1.1$

#### Example 34 🔗

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 $$t= 1$$.

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

$y_p( 1 )\approx -2.0$

#### Example 35 🔗

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 $$t= 4$$.

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

$y_p( 4 )\approx -1.1$

#### Example 36 🔗

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 $$t= 2$$.

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

$y_p( 2 )\approx 1.7$

#### Example 37 🔗

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 $$t= 1$$.

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

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

#### Example 38 🔗

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 $$t= 0$$.

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

$y_p( 0 )\approx 0.00$

#### Example 39 🔗

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 $$t= 3$$.

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

$y_p( 3 )\approx 1.7$

#### Example 40 🔗

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 $$t= 4$$.

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

$y_p( 4 )\approx -1.1$

#### Example 41 🔗

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 $$t= -1$$.

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

$y_p( -1 )\approx 1.0$

#### Example 42 🔗

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 $$t= 4$$.

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

$y_p( 4 )\approx -2.8$

#### Example 43 🔗

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 $$t= 4$$.

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

$y_p( 4 )\approx -1.0$

#### Example 44 🔗

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 $$t= 3$$.

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

$y_p( 3 )\approx -3.9$

#### Example 45 🔗

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 $$t= -4$$.

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

$y_p( -4 )\approx 2.0$

#### Example 46 🔗

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 $$t= -1$$.

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

$y_p( -1 )\approx 0.00$

#### Example 47 🔗

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 $$t= -3$$.

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

$y_p( -3 )\approx 2.0$

#### Example 48 🔗

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 $$t= -4$$.

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

$y_p( -4 )\approx 0.00$

#### Example 49 🔗

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 $$t= -4$$.

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

$y_p( -4 )\approx -1.0$

#### Example 50 🔗

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 $$t= -3$$.

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

$y_p( -3 )\approx -2.0$