Advertisements
Advertisements
Question
Form the differential equation representing the family of curves y = e2x (a + bx), where 'a' and 'b' are arbitrary constants.
Advertisements
Solution
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`.
RELATED QUESTIONS
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 of the family of curves represented by y2 = (x − c)3.
Form the differential equation from the following primitive where constants are arbitrary:
xy = a2
Find the differential equation of the family of curves, x = A cos nt + B sin nt, where A and B are arbitrary constants.
Form the differential equation of the family of curves represented by the equation (a being the parameter):
(2x + a)2 + y2 = a2
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):
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y2 = 4a (x − b)
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Show that y = bex + ce2x is a solution of the differential equation, \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0\]
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:-
\[\left( x + y \right)\frac{dy}{dx} = 1\]
The differential equation which represents the family of curves y = eCx is
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 representing the family of curves y = mx, where m is an arbitrary constant.
The differential equation representing the family of curves y = A sinx + B cosx is ______.
Find the equation of a curve passing through origin and satisfying the differential equation `(1 + x^2) "dy"/"dx" + 2xy` = 4x2
Form the differential equation by eliminating A and B in Ax2 + By2 = 1
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)`.
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.
Find the equation of a curve passing through the point (1, 1). If the tangent drawn at any point P(x, y) on the curve meets the co-ordinate axes at A and B such that P is the mid-point of AB.
The differential equation `y ("d"y)/("d"x) + "c"` represents: ______.
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:
