Expert Answer to "How do I solve the equation dydt=2y−10?"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

daurialebraslmc

Answered question

2022-04-18

How do I solve the equation

\frac{d y}{d t} = 2 y - 10

Answer & Explanation

Cesar Mcfarland

Beginner2022-04-19Added 8 answers

You can employ a method called separation of variables.

Put the entire y to one side and the entire t to the other.
You get:
$\frac{d y}{2 y - 10} = d t$
Now that both sides have been integrated with respect to the relevant variables:
$\int \frac{1}{2 y - 10} d y = \int d t$
$\int \frac{1}{2 (y - 5)} d y = \int d t$
And finally
$\frac{1}{2} \ln (y - 5) = t + c$
You can now write y as follows:
$\ln (y - 5) = 2 t + c$
$y - 5 = c_{1} e^{2 t}$ where $c_{1} = e^{c}$
$y = c_{1} e^{2 t} + 5$
You can substitute back to check your result (calculating $\frac{d y}{d t}$ ) remembering that now it is: $y = c_{1} e^{2 t} + 5$

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