The Problem

The Manipulate and the first Plot of your code don't work because you use a combination of SetDelayed (:=) and ReplaceAll (/.) that doesn't behave like you expected.

When you define

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]]

and now evaluate

g[2, 1, 1, 1]

you get

f0 + 4 (-3 f0 + 3 f1 - ff1) + 8 (2 f0 - 2 f1 + ff1)

instead of the expacted

5

This happens because (due to the use of :=) f[t] on the rhs is first evaluated as f[2] (giving a + 4 c + 8 d) and then the replacement (/. sol[[1]]) is done.

Using Set instead of SetDelayed

If you replace := with =

g2[t_, f0_, f1_, ff1_] = f[t] /. sol[[1]]

then /. will be performed only once, when you define g2 and whenever you use g2 it is replaced by its rhs with the ReplacedAll already performed.

g2[2, 1, 1, 1]

5

Using an additional ReplaceAll

You can also use an additional /. to replace the parameters f0, f1, and ff1 with some values

g3[t_, pf0_, pf1_, pff1_] := f[t] /. sol[[1]] /. {f0 -> pf0, f1 -> pf1, ff1 -> pff1}

now

g3[2, 1, 1, 1]

5

This is similar to the last Plot in your code.

The Problem

The Manipulate and the first Plot of your code don't work because you use a combination of SetDelayed (:=) and ReplaceAll (/.) that doesn't behave like you expected.

When you define

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]]

and now evaluate

g[2, 1, 1, 1]

you get

f0 + 4 (-3 f0 + 3 f1 - ff1) + 8 (2 f0 - 2 f1 + ff1)

instead of the expacted

5

This happens because (due to the use of :=) f[t] on the rhs is first evaluated as f[2] (giving a + 4 c + 8 d) and then the replacement (/. sol[[1]]) is done.

Using Set instead of SetDelayed

If you replace := with =

g2[t_, f0_, f1_, ff1_] = f[t] /. sol[[1]]

then /. will be performed only once, when you define g2 and whenever you use g2 it is replaced by its rhs with the ReplacedAll already performed.

g2[2, 1, 1, 1]

5

Using Evaluate

You can also force the evaluation of the right-hand side by using Evaluate:

g3[t_, f0_, f1_, ff1_] := Evaluate[f[t] /. sol[[1]]]

This way the rhs is evaluated before it is SetDelayed as the definition of g3.

Definition@g3

g3[t_, f0_, f1_, ff1_] := f0 + (-3 f0 + 3 f1 - ff1) t^2 + (2 f0 - 2 f1 + ff1) t^3

Using an additional ReplaceAll

You can also use an additional /. to replace the parameters f0, f1, and ff1 with some values

g4[t_, pf0_, pf1_, pff1_] := f[t] /. sol[[1]] /. {f0 -> pf0, f1 -> pf1, ff1 -> pff1}

now

g4[2, 1, 1, 1]

5

This is similar to the last Plot in your code.

The Problem

The Manipulate and the first Plot of your code don't work because you use a combination of SetDelayed (:=) and ReplaceAll (/.) that doesn't behave like you expected.

When you have defined

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]]

and now evaluate

g[2, 1, 1, 1]

you get

f0 + 4 (-3 f0 + 3 f1 - ff1) + 8 (2 f0 - 2 f1 + ff1)

instead of the expacted

5

This happens because (due to the use of :=) f[t] on the rhs is first evaluated as f[2] (giving a + 4 c + 8 d) and then the replacement (/. sol[[1]]) is done.

Using Set instead of SetDelayed

If you replace := with =

g2[t_, f0_, f1_, ff1_] = f[t] /. sol[[1]]

then /. will be performed only once, when you define g2 and whenever you use g2 it is replaced by its rhs with the ReplacedAll already performed.

g2[2, 1, 1, 1]

5

Using an additional ReplaceAll

You can also use an additional /. to replace the parameters f0, f1, and ff1 with some values

g3[t_, pf0_, pf1_, pff1_] := f[t] /. sol[[1]] /. {f0 -> pf0, f1 -> pf1, ff1 -> pff1}

now

g3[2, 1, 1, 1]

5

This is similar to the last Plot in your code.

The Problem

The Manipulate and the first Plot of your code don't work because you use a combination of SetDelayed (:=) and ReplaceAll (/.) that doesn't behave like you expected.

When you define

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]]

and now evaluate

g[2, 1, 1, 1]

you get

f0 + 4 (-3 f0 + 3 f1 - ff1) + 8 (2 f0 - 2 f1 + ff1)

instead of the expacted

5

This happens because (due to the use of :=) f[t] on the rhs is first evaluated as f[2] (giving a + 4 c + 8 d) and then the replacement (/. sol[[1]]) is done.

Using Set instead of SetDelayed

If you replace := with =

g2[t_, f0_, f1_, ff1_] = f[t] /. sol[[1]]

then /. will be performed only once, when you define g2 and whenever you use g2 it is replaced by its rhs with the ReplacedAll already performed.

g2[2, 1, 1, 1]

5

Using an additional ReplaceAll

You can also use an additional /. to replace the parameters f0, f1, and ff1 with some values

g3[t_, pf0_, pf1_, pff1_] := f[t] /. sol[[1]] /. {f0 -> pf0, f1 -> pf1, ff1 -> pff1}

now

g3[2, 1, 1, 1]

5

This is similar to the last Plot in your code.

The Manipulate and the first Plot of your code don't work because you use a combination of SetDelayed (:=) and ReplaceAll (/.) that doesn't behave like you expected.

When you have defined

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]]

and now evaluate

g[2, 1, 1, 1]

you get

f0 + 4 (-3 f0 + 3 f1 - ff1) + 8 (2 f0 - 2 f1 + ff1)

instead of the expacted

5

This happens because (due to the use of :=) f[t] on the rhs is evaluated as f[2] (giving a + 4 c + 8 d) and then the replacement (/. sol[[1]]) is done.

The Problem

The Manipulate and the first Plot of your code don't work because you use a combination of SetDelayed (:=) and ReplaceAll (/.) that doesn't behave like you expected.

When you have defined

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]]

and now evaluate

g[2, 1, 1, 1]

you get

f0 + 4 (-3 f0 + 3 f1 - ff1) + 8 (2 f0 - 2 f1 + ff1)

instead of the expacted

5

This happens because (due to the use of :=) f[t] on the rhs is first evaluated as f[2] (giving a + 4 c + 8 d) and then the replacement (/. sol[[1]]) is done.

Using Set instead of SetDelayed

If you replace := with =

g2[t_, f0_, f1_, ff1_] = f[t] /. sol[[1]]

then /. will be performed only once, when you define g2 and whenever you use g2 it is replaced by its rhs with the ReplacedAll already performed.

g2[2, 1, 1, 1]

5

Using an additional ReplaceAll

You can also use an additional /. to replace the parameters f0, f1, and ff1 with some values

g3[t_, pf0_, pf1_, pff1_] := f[t] /. sol[[1]] /. {f0 -> pf0, f1 -> pf1, ff1 -> pff1}

now

g3[2, 1, 1, 1]

5

This is similar to the last Plot in your code.