Question

A manufacturing company has found that the cost C of operating and maintaining the equipment is related to the length ’m’ of intervals between overhauls by the equation `"m"^2 "dC"/"dm" + 2"mC"` = 2 and c = 4 and when  = 2. Find the relationship between C and m

Answer 1

`"m"^2 "dC"/"dm" + 2"mC"` = 2

÷ each term by m² 

`"c"/"dm" + (2"mc")/"m"^2 = 2/"m"^2`

`"dc"/"dm" + (2"c")/"m" = 2/"m"^2`

This is a first order linear differential equation of the form

`"dc"/"dm" + "Pc"` = Q 

Where P = `2"m"` and Q = `2/"m"^2`

`int "Pdm" = 2 int 1/"m" "dm" =2 log "m" = log "m"^2`

I.F = `"e"^(int"Pdm")`

= `"e"^(log"m"^2)`

= m²

General solution is

 C(I.F) = `int  "Q" xx ("I.F")  "dm" + "k"`

C(m²) = `int 2/"m"^2 xx ("m"^2)  "dm" + "k"`

C(m²) = `int 2"dm" + "k"`

Cm² = 2m + k  .......(1)

When C = 4 and m = 2, we have

(4)(2)² = 2(2) + k

16 = 4 + k = 12

Equation (1)

Cm² = 2m + 12

Cm² = 2(m + 6)