Advertisements
Advertisements
Question
If F is the constant force generated by the motor of an automobile of mass M, its velocity V is given by `"M""dv"/"dt"` = F – kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0
Advertisements
Solution
Given equation `"M" "dV"/"dt"` = F – kV ......(∵ F and k are constant)
`"M" "dV"/"dt" = "k"("F"/"k" - "V")`
`int "dV"/(("F"/"k" - "V")) = "k"/"M" int "dt"`
`- log ("F"/"k" - "V") = "k"/"M" "t" + "c"` .......(1)
Given V = 0 and t = 0
⇒ `- log "F"/"k"` = c
Substituting in (1)
`- log ("F"/"k" - "V") = "kt"/"M" - log ("F"/"k")`
`log("F"/"k") - log (("F"- "Vk")/"k") = "kt"/"M"`
`log (("F"/"k")/(("F" - "Vk")/"k")) = "kt"/"M"`
`("F"/("F" - "Vk")) = "e"^("kt"/"M")`
F = `("F" - "Vk") "e"^("kt"/"M")`
APPEARS IN
RELATED QUESTIONS
Solve the following differential equation:
`("d"y)/("d"x) = "e"^(x + y) - x^3"e"^y`
Solve the following differential equation:
`2xy"d"x + (x^2 + 2y^2)"d"y` = 0
Choose the correct alternative:
The solution of `("d"y)/("d"x) = 2^(y - x)` is
Choose the correct alternative:
If sin x is the integrating factor of the linear differential equation `("d"y)/("d"x) + "P"y = "Q"`, then P is
Choose the correct alternative:
The number of arbitrary constants in the general solutions of order n and n +1are respectively
Solve: `("d"y)/("d"x) = "ae"^y`
Solve: `(1 + x^2)/(1 + y) = xy ("d"y)/("d"x)`
Solve: ydx – xdy = 0 dy
Find the curve whose gradient at any point P(x, y) on it is `(x - "a")/(y - "b")` and which passes through the origin
Solve the following homogeneous differential equation:
(y2 – 2xy) dx = (x2 – 2xy) dy
Solve the following homogeneous differential equation:
An electric manufacturing company makes small household switches. The company estimates the marginal revenue function for these switches to be (x2 + y2) dy = xy dx where x represents the number of units (in thousands). What is the total revenue function?
Solve the following:
`("d"y)/("d"x) + (3x^2)/(1 + x^3) y = (1 + x^2)/(1 + x^3)`
Choose the correct alternative:
If y = ex + c – c3 then its differential equation is
Choose the correct alternative:
If sec2 x is an integrating factor of the differential equation `("d"y)/("d"x) + "P"y` = Q then P =
Choose the correct alternative:
Which of the following is the homogeneous differential equation?
Choose the correct alternative:
The solution of the differential equation `("d"y)/("d"x) = y/x + (f(y/x))/(f"'"(y/x))` is
Solve `x ("d"y)/(d"x) + 2y = x^4`
Solve `("d"y)/("d"x) + y cos x + x = 2 cos x`
