Advertisements
Advertisements
Question
A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
Advertisements
Solution
Let S be the displacement of the particle at time 't'.
Its velocity and acceleration are `(dS)/dt and (d^2S)/dt^2` respectively.
Here `(d^2S)/dt^2 ∝ (dS)/dt`
⇒ `(d^2S)/dt^2 = k(dS)/dt`, ...(where k is constant ≠ 0)
This is the required differential equation.
APPEARS IN
RELATED QUESTIONS
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
Ax2 + By2 = 1
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = Ae5x + Be-5x
Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.
In the following example verify that the given expression is a solution of the corresponding differential equation:
xy = log y +c; `"dy"/"dx" = "y"^2/(1 - "xy")`
Solve the following differential equation:
`"dy"/"dx" = - "k",` where k is a constant.
Solve the following differential equation:
`(cos^2y)/x dy + (cos^2x)/y dx` = 0
For the following differential equation find the particular solution satisfying the given condition:
`(e^y + 1) cos x + e^y sin x. dy/dx = 0, "when" x = pi/6,` y = 0
Reduce the following differential equation to the variable separable form and hence solve:
`"dy"/"dx" = cos("x + y")`
Reduce the following differential equation to the variable separable form and hence solve:
`("x - y")^2 "dy"/"dx" = "a"^2`
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
The particular solution of `dy/dx = xe^(y - x)`, when x = y = 0 is ______.
In the following example verify that the given function is a solution of the differential equation.
`"y" = 3 "cos" (log "x") + 4 sin (log "x"); "x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola `"x"^2/16 - "y"^2/36 = "k"`.
Solve the following differential equation:
`"dy"/"dx" = "x"^2"y" + "y"`
Solve the following differential equation:
`"dy"/"dx" = ("2y" - "x")/("2y + x")`
Solve the following differential equation:
x dy = (x + y + 1) dx
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
Find the particular solution of the following differential equation:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
Find the particular solution of the following differential equation:
(x + y)dy + (x - y)dx = 0; when x = 1 = y
Select and write the correct alternative from the given option for the question
Solution of the equation `x ("d"y)/("d"x)` = y log y is
Select and write the correct alternative from the given option for the question
General solution of `y - x ("d"y)/("d"x)` = 0 is
Select and write the correct alternative from the given option for the question
The solution of `("d"y)/("d"x)` = 1 is
Find the differential equation of family of lines making equal intercepts on coordinate axes
Find the general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
Find the differential equation from the relation x2 + 4y2 = 4b2
Find the differential equation of the family of all non-horizontal lines in a plane
Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis
The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is ______.
For the curve C: (x2 + y2 – 3) + (x2 – y2 – 1)5 = 0, the value of 3y' – y3 y", at the point (α, α), α < 0, on C, is equal to ______.
The differential equation representing the family of ellipse having foci either on the x-axis or on the y-axis centre at the origin and passing through the point (0, 3) is ______.
The differential equation of all parabolas whose axis is Y-axis, is ______.
The differential equation of the family of circles touching Y-axis at the origin is ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
Solve the differential equation
cos2(x – 2y) = `1 - 2dy/dx`
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
Form the differential equation of all concentric circles having centre at the origin.
The differential equation whose solution represents the family \[x^{2}y=4e^{x}+c\], where c is an arbitrary constant, is
