# Symbolic differentiation [closed]

I have two symbolic differential equations and I want to used them in another equation as follows:

eq1 = Dt[Ai] == mt*δfm + 2*Ω^2;
eq2 = Dt[Ri] == h*δmm + 2*Ω^2;
ya = Ai*Ri;
Dt[ya]


Ri Dt[Ai] + Ai Dt[Ri]

Instead, I want output that inserts Dt[Ai] and Dt[Ri] and gives the simplified form.

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• Do you not get error messages from your code? – Michael E2 Jan 7 '16 at 1:24
• Perhaps ya = Ai Ri; Simplify[Dt[ya] /. {Dt[Ai]->h δfm + 2 Ω^2, Dt[Ri]->mt δmm + 2 Ω^2}] – Bill Jan 7 '16 at 1:44
• Hi i mistakenly posted the wrong equations. I have updated the post. – aly Jan 7 '16 at 2:18
• @bill & Michael. Thanks for your quick answer. I want a solution with the way i have defined because i will need those differential equation several times in many equations and following the routine you recommended will be bulky when used every time. – aly Jan 7 '16 at 2:30

## 3 Answers

If you want teach Mathematica to make the substitution automatically, you should define special evaluation rules for Dt[Ai] and Dt[Ri] with UpSet (^=).

Dt[Ai] ^= mt*\[Delta]fm + 2*\[CapitalOmega]^2;
Dt[Ri] ^= h*\[Delta]mm + 2*\[CapitalOmega]^2;
ya = Ai Ri;
Dt[ya]


Ri (mt δfm + 2 Ω^2) + Ai (h δmm + 2 Ω^2)

You can also replace the head of your equations, turning them temporarily into Rule objects for the purpose of the substitution.

eq1 = Dt[Ai] == mt*δfm + 2*Ω^2;
eq2 = Dt[Ri] == h*δmm + 2*Ω^2;
ya = Ai*Ri;
Dt[ya]

Dt[ya] /. Rule @@@ {eq1, eq2}

(* Ri Dt[Ai] + Ai Dt[Ri] *)

(* Ri (mt δfm + 2 Ω^2) + Ai (h δmm + 2 Ω^2) *)


I got the solution that satisfies me:

eq1 = Dt[Ai] == mt*δfm + 2*Ω^2
eq2 = Dt[Ri] == h*δmm + 2*Ω^2;
ya = Ai*Ri;

result = Simplify[Dt[ya] /.{Dt[Ai] -> eq1, Dt[Ri] -> eq2}]


I am satisfied with the solution but if you have a better solution, please share. Thanks to Bill as I derived the solution following his answer.

• This gives result in form of __ == __ == __, is this really what you expect? – Kuba Jan 8 '16 at 7:59
• It gives me the correct results – aly Jan 30 '16 at 13:38