Question

Find the solution to the differential equation

dB/dt+4B=20

with B(1)=30

Answer 1

Solution:
`B=5+25e^(4-4t)`

Explanation:

Given:
`(dB)/(dt)+4B=20`

with B(1)=30

The differential equation in form of linear differential equation,

`(dy)/(dt)+Py=Q`

Integral factor, IF:
`e^(\int Pdt)`

General Solution:

`y\cdot IF=\int Q\cdot IFdt`

`(dB)/(dt)+4B=20`

P=4, Q=20

IF=
`e^(\int 4dt)=e^(4t)`

Solution:

`Be^(4t)=\int 20e^(4t)dt`

`Be^(4t)=5e^(4t)+C`

`B=5+Ce^(-4t)`

B(1)=30 , Put t=1, B=30

`30=5+Ce^(-4)`

`C=25e^4`

`B=5+25e^(4-4t)`