Advertisements
Advertisements
प्रश्न
Form the differential equation representing the family of curves `y2 = m(a2 - x2) by eliminating the arbitrary constants 'm' and 'a'.
Advertisements
उत्तर
The equation y2 = m(a2 - x2) where m and a are arbitrary constants.
y2 = m(a2 - x2) ......(i)
Differentiate (i) w.r.t.x.
`2"y"(d"y")/(d"x")` = -2mx ...(ii)
⇒ -2m = `2 ("y")/("x") (d"y")/(d"x")`
Differentiate (ii) w.r.t.x.
`2["y" (d^2"y")/(d"x"^2) + ((d"y")/(d"x"))^2] ` = -2m .....(iii)
From (ii) and (iii), we get
`2["y" (d^2"y")/(d"x"^2) + ((d"y")/(d"x"))^2] = 2 ("y")/("x") (d"y")/(d"x")`
`"y"(d^2"y")/(d"x"^2) + ((d"y")/(d"x"))^2-("y"/"x")(d"y")/(d"x")` = 0
therefore the required differential equation is `"y"(d^2"y")/(d"x"^2) + ((d"y")/(d"x"))^2-("y"/"x")(d"y")/(d"x")` = 0
संबंधित प्रश्न
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
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:
xy = a2
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):
(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):
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
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:-
\[\frac{dy}{dx} - \frac{2xy}{1 + x^2} = x^2 + 2\]
Find one-parameter families of solution curves of the following differential equation:-
\[\left( x + y \right)\frac{dy}{dx} = 1\]
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:-
\[x \log x\frac{dy}{dx} + y = 2 \log x\]
Write the differential equation representing family of curves y = mx, where m is arbitrary constant.
Form the differential equation of the family of ellipses having foci on y-axis and centre at the origin.
Form the differential equation representing the family of curves y = e2x (a + bx), where 'a' and 'b' are arbitrary constants.
Form the differential equation representing the family of curves y = A sin x, by eliminating the arbitrary constant A.
Find the differential equation of the family of lines through the origin.
Find the equation of a curve whose tangent at any point on it, different from origin, has slope `y + y/x`.
The solution of the differential equation `2x * "dy"/"dx" y` = 3 represents a family of ______.
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 curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is ______.
The differential equation representing the family of circles x2 + (y – a)2 = a2 will be of order two.
Find the equation of the curve at every point of which the tangent line has a slope of 2x:
Form the differential equation of the family of hyperbola having foci on x-axis and centre at origin.
