Advertisements
Advertisements
प्रश्न
The differential equation of the family of curves y2 = 4a(x + a) is ______.
पर्याय
`y^2 - 4 ("d"y)/("d"x)(x + ("d"y)/("d"x))`
`2y ("d"y)/("d"x)` = 4a
`y ("d"^2y)/("d"x^2) + (("d"y)/("d"x))^2` = 0
`2x ("d"y)/("d"x) + y(("d"y)/("d"x))^2 - y`
उत्तर
The differential equation of the family of curves y2 = 4a(x + a) is `2x ("d"y)/("d"x) + y(("d"y)/("d"x))^2 - y`.
Explanation:
The given equation of family of curves is y2 = 4a(x + a)
⇒ y2 = 4ax + 4a2 .......(1)
Differentiating both sides, w.r.t. x, we get
`2y * ("d"y)/("d"x)` = 4a
⇒ `y * ("d"y)/("d"x)` = 2a
⇒ `y/2 ("d"y)/("d"x)` = a
Now, putting the value of a in equation (1) we get
`y^2 = 4x(y/2 ("d"y)/("d"x)) + 4(y/2 * ("d"y)/("d"x))^2`
⇒ `y^2 = 2xy ("d"y)/("d"x) + y^2 (("d"y)/("d"x))^2`
⇒ y = `2x ("d"y)/("d"x) + y(("d"y)/("d"x))^2`
⇒ `2x * ("d"y)/("d"x) + y * (("d"y)/("d"x))^2 - y` = 0
APPEARS IN
संबंधित प्रश्न
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 from the following primitive where constants are arbitrary:
y = ax2 + bx + c
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):
x2 − 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)
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + y2 = ax3
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = eax
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:-
\[\frac{dy}{dx} - y = \cos 2x\]
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:-
\[\frac{dy}{dx} \cos^2 x = \tan x - y\]
Form the differential equation of the family of ellipses having foci on y-axis and centre at the origin.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Find the area of the region bounded by the curves (x -1)2 + y2 = 1 and x2 + y2 = 1, using integration.
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.
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)`
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 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
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.
