Pregunta
Solve the differential equation
for a constant growth rate
.
Ask by Wyatt Ross. in South Africa
Dec 09,2024
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The general solution to the differential equation
is
, where
is a constant.
Solución
To solve the differential equation
we can use the method of separation of variables. We start by rearranging the equation to separate the variables
and
:
Next, we integrate both sides. The left side integrates to
, and the right side integrates to
:
where
is the constant of integration.
To solve for
, we exponentiate both sides:
Let
, which is a positive constant. Thus, we can write:
Since
is positive, we can drop the absolute value by allowing
to take on any real value (positive or negative):
Finally, we can express the solution in a more general form by letting
(which can be any real number):
where
is a constant determined by initial conditions. This is the general solution to the differential equation
.
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Como
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Bonus Knowledge
To solve the differential equation
, we start by separating the variables. This involves rewriting the equation as:
Next, we integrate both sides. The left side with respect to
gives us:
And integrating the right side with respect to
gives:
Now, combining these results and simplifying leads to:
where
. To eliminate the logarithm, we exponentiate both sides:
Letting
, which is a positive constant, we have:
Thus, the solution to the differential equation
is:
where
is determined by initial conditions if provided.

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