Advertisements
Advertisements
प्रश्न
Form the differential equation from the following primitive where constants are arbitrary:
y = cx + 2c2 + c3
Advertisements
उत्तर
The equation of family of curves is \[y = cx + 2 c^2 + c^3..............(1)\]
Where `c` is an arbitrary constant.
This equation contains only one arbitrary constant, so we shall get a differential equation of first order.
Differentiating equation (1) with respect to x, we get
\[\frac{dy}{dx} = c...............(2)\]
Putting the value of `c` in equation (1), we get
\[y = x\frac{dy}{dx} + 2 \left( \frac{dy}{dx} \right)^2 + \left( \frac{dy}{dx} \right)^3\]
It is the required differential equation.
APPEARS IN
संबंधित प्रश्न
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
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`
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.
Form the differential equation corresponding to y = emx by eliminating m.
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):
(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):
y2 = 4a (x − b)
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = ax3
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
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x\]
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:-
\[\frac{dy}{dx} - y = \cos 2x\]
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:-
\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]
Find one-parameter families of solution curves of the following differential equation:-
\[e^{- y} \sec^2 y dy = dx + x dy\]
Find one-parameter families of solution curves of the following differential equation:-
\[x \log x\frac{dy}{dx} + y = 2 \log x\]
Find one-parameter families of solution curves of the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
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 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.
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.
The solution of the differential equation `2x * "dy"/"dx" y` = 3 represents a family of ______.
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 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 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 of the family of curves y2 = 4a(x + a) is ______.
Find the equation of the curve at every point of which the tangent line has a slope of 2x:
