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Putting a differential equation into a certain form

Posted 1 month ago
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Hi guys so I have this differential equation

DSolve[{M'[t] == 1/4500 (660 - M[t]) M[t] - 10, M[0] == 100}, M[t], t] 

which gives me a result

(300 (4 + Sqrt[71] - 4 E^((Sqrt[71] t)/75) + 
Sqrt[71] E^((Sqrt[71] t)/75)))/(23 + 3 Sqrt[71] - 
23 E^((Sqrt[71] t)/75) + 3 Sqrt[71] E^((Sqrt[71] t)/75))

that I need to get into the form (150 (b + a E^((Sqrt[71] t)/75)))/(c + 55 E^((Sqrt[71] t)/75)) and I need to find a,b,c I have tried solve always but no luck is there any command that I can use to find a,b,c?

2 Replies

Simplify does it already. However, you can use SolveAlways by setting the exponential to a new symbol:

sol = DSolveValue[{M'[t] == 1/4500 (660 - M[t]) M[t] - 10, 
    M[0] == 100}, M[t], t];
preferredForm = (150 (b + a E^((Sqrt[71] t)/75)))/(c + 
     55 E^((Sqrt[71] t)/75));
SolveAlways[preferredForm == sol /. E^((Sqrt[71] t)/75) -> x, x]
sol /. %[[1]]
Posted 1 month ago

Thank you so much that worked.

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