Mathematica Asked by Sosa on May 8, 2021

I’m new at coding and I got this simple differential equiation and the plot of this function involved, I can obtain the value of the dependant variable based on the independant varible as usual, what I want to do now is to find a new value of the independant variable giving it the dependant variable based on the ploted graph, in other words I want to find y for a given x from the graph. this is my short code. Only half of it works.

```
Sol = NDSolve[{a'[t] == Sqrt[a[t]], a[0] == 3}, a, {t, 0, 180}]
graph = Plot[a[t] /. Sol, {t, 0, 180}]
FindRoot[a[t] == 0.44 /. graph, {t, 0}] (*this is the part that doesn't work*)
a[t] /. Sol /. t -> 180
```

Any kind of help would be hugely appreciated. Thanks c:

First, you should not use variable names with capital letters, those are used by the system. I corrected your syntax. Here is the working code. Note that the `FindRoot`

gives warnings because the specified value is out of bounds:

```
sol[t_] =
a[t] /. NDSolve[{a'[t] == Sqrt[a[t]], a[0] == 3}, a, {t, 0, 180}][[1]]
graph = Plot[sol[t], {t, 0, 180}]
FindRoot[sol[t] == 0.4, {t, 0}]
sol[180]
```

Answered by Daniel Huber on May 8, 2021

Another way is use `WhenEvent`

and `Reap`

and `Sow`

.(Also need to replace `0.44`

to another number such as `10`

).

```
{{sol}, {pts}} =
NDSolve[{a'[t] == Sqrt[a[t]], a[0] == 3,
WhenEvent[a[t] == 10, {Sow[t]}]}, a, {t, 0, 180}] // Reap
Plot[a[t] /. First@sol // Evaluate, {t, 0, 10},
Epilog -> {PointSize[Large], Red,
Point[({#, a[#]} /. sol) & /@ pts]}]
```

`{{2.86045}}`

**Appendix**

The questioner's original approach.

```
sol = NDSolve[{a'[t] == Sqrt[a[t]], a[0] == 3}, a, {t, 0, 180}]
graph = Plot[a[t] /. sol, {t, 0, 180}]
FindRoot[a[t] == 10 /. sol, {t,
0}] (*this is the part that doesn't work*)
a[t] /. sol /. %
```

`{t -> 2.86045}`

`{10.}`

Answered by cvgmt on May 8, 2021

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