Advertisements
Advertisements
प्रश्न
The differential equation of the family of curves x2 + y2 – 2ay = 0, where a is arbitrary constant, is ______.
पर्याय
`(x^2 - y^2) ("d"y)/("d"x)` = 2xy
`2(x^2 + y^2) ("d"y)/("d"x)` = xy
`2(x^2 - y^2) ("d"y)/("d"x)` = xy
`(x^2 + y^2) ("d"y)/("d"x)` = 2xy
उत्तर
The differential equation of the family of curves x2 + y2 – 2ay = 0, where a is arbitrary constant, is `(x^2 - y^2) ("d"y)/("d"x)` = 2xy.
Explanation:
The given equation is x2 + y2 – 2ay = 0 ......(1)
Differentiating w.r.t. x, we have
`2x + 2y * ("d"y)/("d"x) - 2"a" ("d"y)/("d"x)` = 0
⇒ `x + y ("d"y)/("d"x) - "a" ("d"y)/("d"x)` = 0
⇒ `x + (y - "a") ("d"y)/("d"x)` = 0
⇒ `(y - "a") ("d"y)/("d"x)` = – x
⇒ y – a = `(-x)/(("d"y)/("d"x))`
⇒ a = `y + x/(("d"y)/("d"x))`
⇒ a = `(y * ("d"y)/("d"x) + x)/(("d"y)/("d"x))`
Putting the value of a in equation (1) we get
`x^2 + y^2 - 2y [(y ("d"y)/("d"x) + x)/(("d"y)/("d"x))]` = 0
⇒ `(x^2 + y^2) ("d"y)/("d"x) - 2y(y ("d"y)/("d"x) + x)` = 0
⇒ `(x^2 + y^2) ("d"y)/("d"x) - 2y^2 ("d"y)/("d"x) - 2xy` = 0
⇒ `(x^2 + y^2 - 2y^2) ("d"y)/("d"x^2)` = 2xy
⇒ `(x^2 - y^2) ("d"y)/("d"x)` = 2xy
APPEARS IN
संबंधित प्रश्न
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
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 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 circles in the first quadrant which touch the coordinate axes.
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 corresponding to y = emx by eliminating m.
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.
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):
y2 = 4a (x − b)
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = eax
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:-
\[\left( x + y \right)\frac{dy}{dx} = 1\]
Write the differential equation representing family of curves y = mx, where m is arbitrary constant.
Write the order of the differential equation representing the family of curves y = ax + a3.
The differential equation which represents the family of curves y = eCx is
Find the differential equation of the family of lines through the origin.
Find the differential equation of system of concentric circles with centre (1, 2).
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 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 of the family of curves y2 = 4a(x + a) is ______.
From the differential equation of the family of circles touching the y-axis at origin
Form the differential equation of the family of hyperbola having foci on x-axis and centre at origin.
