Advertisements
Advertisements
प्रश्न
Find the equation of a curve passing through origin if the slope of the tangent to the curve at any point (x, y) is equal to the square of the difference of the abcissa and ordinate of the point.
Advertisements
उत्तर
Here, slope of the tangent of the curve = `("d"y)/("d"x)` and the difference between the abscissa and ordinate = x – y.
∴ As per the condition, `("d"y)/("d"x) = (x - y)^2`
Put x – y = v
`1 - ("d"y)/("d"x) = "dv"/("d"x)`
∴ `("d"y)/("d"x) = 1 - "dv"/"dx"`
∴ The equation becomes `1 - "dv"/"dx" = "v"^2`
⇒ `"dv"/"dx" = 1 - "v"^2`
⇒ `"dv"/(1 - "v"^2)` = dx
Integrating both sides, we get
`int "dv"/(1 - "v"^2) = int "d"x`
⇒ `1/2 log |(1 + "v")/(1 - "v")|` = x + c
⇒ `1/2 log|(1 + x - y)/(1 - x + y)|` = x + c ......(1)
Since, the curve is passing through (0, 0)
Then `1/2 log|(1 + 0 - 0)/(1 - 0 + 0)|` = 0 + c
⇒ c = 0
∴ On putting c = 0 in equation (1) we get
`1/2 log |(1 + x - y)/(1 - x + y)|` = x
⇒ `log|(1 + x - y)/(1 - x + y)|` = 2x
∴ `(1 + x - y)/(1 - x + y)|` = e2x
⇒ (1 + x – y) = e2x (1 – x + y)
Hence, the required equation is (1 + x – y) = e2x (1 – x + y).
APPEARS IN
संबंधित प्रश्न
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Which of the following differential equation has y = x as one of its particular solution?
A. `(d^2y)/(dx^2) - x^2 (dy)/(dx) + xy = x`
B. `(d^2y)/(dx^2) + x dy/dx + xy = x`
C. `(d^2y)/(dx^2) - x^2 dy/dx + xy = 0`
D. `(d^2y)/(dx^2) + x dy/dx + xy = 0`
For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3
Form the differential equation from the following primitive where constants are arbitrary:
xy = a2
Form the differential equation from the following primitive where constants are arbitrary:
y = ax2 + bx + c
Find the differential equation of the family of curves y = Ae2x + Be−2x, where A and B are arbitrary constants.
Form the differential equation corresponding to y2 = a (b − x2) by eliminating a and b.
Form the differential equation corresponding to y2 − 2ay + x2 = a2 by eliminating a.
Form the differential equation corresponding to (x − a)2 + (y − b)2 = r2 by eliminating a and b.
Form the differential equation of the family of curves represented by the equation (a being the parameter):
(2x + a)2 + y2 = a2
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + (y − b)2 = 1
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
(x − a)2 − y2 = 1
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y2 = 4a (x − b)
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = ax3
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + y2 = ax3
Find one-parameter families of solution curves of the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{mx}\], m is a given real number.
Find one-parameter families of solution curves of the following differential equation:-
\[\frac{dy}{dx} \cos^2 x = \tan x - y\]
Find one-parameter families of solution curves of the following differential equation:-
\[e^{- y} \sec^2 y dy = dx + x dy\]
Write the differential equation representing family of curves y = mx, where m is arbitrary constant.
Form the differential equation of the family of ellipses having foci on y-axis and centre at the origin.
Form the differential equation representing the family of curves `y2 = m(a2 - x2) by eliminating the arbitrary constants 'm' and 'a'.
Find the differential equation of the family of curves y = Ae2x + B.e–2x.
Find the equation of a curve passing through the point (1, 1) if the perpendicular distance of the origin from the normal at any point P(x, y) of the curve is equal to the distance of P from the x-axis.
Find the equation of a curve passing through origin and satisfying the differential equation `(1 + x^2) "dy"/"dx" + 2xy` = 4x2
Find the equation of a curve passing through (2, 1) if the slope of the tangent to the curve at any point (x, y) is `(x^2 + y^2)/(2xy)`.
Family y = Ax + A3 of curves is represented by the differential equation of degree ______.
The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is ______.
The differential equation of the family of curves y2 = 4a(x + a) is ______.
Differential equation representing the family of curves y = ex (Acosx + Bsinx) is `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y` = 0
The area above the x-axis and under the curve `y = sqrt(1/x - 1)` for `1/2 ≤ x ≤ 1` is:
From the differential equation of the family of circles touching the y-axis at origin
