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 ellipses having foci on y-axis and centre at origin.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.
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`
Form the differential equation of the family of curves represented by y2 = (x − c)3.
Form the differential equation corresponding to y = emx by eliminating m.
Form the differential equation from the following primitive where constants are arbitrary:
y = ax2 + bx + c
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):
y2 = 4a (x − b)
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x\]
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:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x}\]
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:-
\[x \log x\frac{dy}{dx} + y = 2 \log x\]
The family of curves in which the sub tangent at any point of a curve is double the abscissae, is given by
Form the differential equation of the family of ellipses having foci on y-axis and centre at the origin.
Find the differential equation of the family of curves y = Ae2x + B.e–2x.
Find the equation of a curve whose tangent at any point on it, different from origin, has slope `y + y/x`.
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.
The differential equation representing the family of curves y = A sinx + B cosx is ______.
Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve at any point (x, y) is `(y - 1)/(x^2 + x)`
The differential equation of the family of curves x2 + y2 – 2ay = 0, where a is arbitrary constant, is ______.
The differential equation representing the family of circles x2 + (y – a)2 = a2 will be of order two.
From the differential equation of the family of circles touching the y-axis at origin
