Advertisements
Advertisements
Question
Find the equation of a curve whose tangent at any point on it, different from origin, has slope `y + y/x`.
Solution
Given `"dy"/"dx" = y + y/x`
= `"y"(1 + 1/x)`
⇒ `"dy"/y = (1 + 1/x)"d"x`
Integrating both sides, we get
logy = x + logx + c
⇒ `log(y/x)` = x + c
⇒ `y/x = "e"^(x + "c") `
= `"e"^x * "e"^"c"`
⇒ `y/x` = k . ex
⇒ y = kx . ex.
APPEARS IN
RELATED QUESTIONS
Which of the following differential equations has y = c1 ex + c2 e–x as the general solution?
(A) `(d^2y)/(dx^2) + y = 0`
(B) `(d^2y)/(dx^2) - y = 0`
(C) `(d^2y)/(dx^2) + 1 = 0`
(D) `(d^2y)/(dx^2) - 1 = 0`
Show that the family of curves for which `dy/dx = (x^2+y^2)/(2x^2)` is given by x2 - y2 = cx
Form the differential equation from the following primitive where constants are arbitrary:
y2 = 4ax
Find the differential equation of the family of curves y = Ae2x + Be−2x, where A and B are arbitrary constants.
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 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 of the family of curves represented by the equation (a being the parameter):
(x − a)2 + 2y2 = 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):
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = eax
For the differential equation xy \[\frac{dy}{dx}\] = (x + 2) (y + 2). Find the solution curve passing through the point (1, −1).
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:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x}\]
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 lines through the origin.
The solution of the differential equation `2x * "dy"/"dx" y` = 3 represents a family of ______.
The differential equation representing the family of curves y = A sinx + B cosx is ______.
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
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: ______.
The differential equation representing the family of circles x2 + (y – a)2 = a2 will be of order two.
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:
Form the differential equation of family of circles having centre on y-axis and raduis 3 units
