Advertisements
Advertisements
प्रश्न
Form the differential equation representing the family of curves y = e2x (a + bx), where 'a' and 'b' are arbitrary constants.
Advertisements
उत्तर
Given: y = e2x (a + bx)
Differentiating the above equation, we get
`(dy)/(dx) = be^(2x) + 2 (a + bx)e^(2x)`
`= (dy)/(dx) = be^(2x) + 2y ...("i") [∵ y = e^(2x) (a + bx)]`
differentiating the above equation, we get
`(d^2y)/(dx^2) = 2 be^(2x) + 2(dy)/(dx)`
= `(d^2y)/(dx^2) = 2 ((dy)/(dx) - 2y) + 2(dy)/(dx) ...[∵ "from" ("i") "we get", be^(2x) = (dy)/(dx) - 2y]`
= `(d^2y)/(dx^2) = 4(dy)/(dx)- 4y`
Hence, the required differential equation is `(d^2y)/(dx^2) - 4 (dy)/(dx) + 4y= 0`.
संबंधित प्रश्न
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.
Form the differential equation representing the family of curves given by (x – a)2 + 2y2 = a2, where a is an arbitrary constant.
Form the differential equation of the family of curves represented by y2 = (x − c)3.
Form the differential equation from the following primitive where constants are arbitrary:
y = cx + 2c2 + c3
Form the differential equation from the following primitive where constants are arbitrary:
xy = a2
Form the differential equation corresponding to y2 − 2ay + x2 = a2 by eliminating a.
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):
(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):
x2 + y2 = ax3
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Find one-parameter families of solution curves of the following differential equation:-
\[\frac{dy}{dx} + y \cos x = e^{\sin x} \cos x\]
Find one-parameter families of solution curves of the following differential equation:-
\[\frac{dy}{dx} \cos^2 x = \tan x - y\]
Write the differential equation representing family of curves y = mx, where m is arbitrary constant.
Form the differential equation representing the family of curves y = mx, where m is an arbitrary constant.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Form the differential equation representing the family of curves y = A sin x, by eliminating the arbitrary constant A.
Find the equation of a curve whose tangent at any point on it, different from origin, has slope `y + y/x`.
Form the differential equation of all circles which pass through origin and whose centres lie on y-axis.
Form the differential equation by eliminating A and B in Ax2 + By2 = 1
Family y = Ax + A3 of curves is represented by the differential equation of degree ______.
The differential equation `y ("d"y)/("d"x) + "c"` represents: ______.
The differential equation of the family of curves x2 + y2 – 2ay = 0, where a is arbitrary constant, is ______.
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 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
Form the differential equation of family of circles having centre on y-axis and raduis 3 units
