Solve this separable differential equation How would I go about solving the following separable differential equation? (dx(t))/(dt)=8-3x with x(0)=4?

suchonosdy 2022-07-17 Answered
Solve this separable differential equation
How would I go about solving the following separable differential equation?
d x ( t ) d t = 8 3 x with x ( 0 ) = 4?
My solution thus far is the following:
d x ( t ) 8 3 x = d t + C
ln | 8 3 x ( t ) | = 3 ( t + C )
Now using the fact that x ( 0 ) = 4 we can solve for C = 1 3 ln | 4 | . From this, I would seem to get
| 8 3 x ( t ) | = e 3 t 1 3 ln 4
but something seems to go wrong. Am I making a mistake somewhere and how could I solve this entirely?
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Answers (1)

taguetzbo
Answered 2022-07-18 Author has 16 answers
d x ( t ) d t = 8 3 x
d x ( t ) 8 3 x = d t
l n ( | 8 3 x | ) 3 = t + C
( l n | 8 3 x | ) = 3 t + 3 C
When t=0, x=4 therefore
l n 4 = 3 C and C = l n 4 3
( l n | 8 3 x | ) = 3 t + l n 4
| 8 3 x | = e 3 t + l n 4 = e 3 t . e l n 4 = 4. e 3 t
therefore required solution
| 8 3 x | = 4. e 3 t
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