Advertisements
Advertisements
Question
Differential equation representing the family of curves y = ex (Acosx + Bsinx) is `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y` = 0
Options
True
False
Advertisements
Solution
This statement is True.
Explanation:
Given equation is y = ex (Acosx + Bsinx)
Differentiating both sides, we get
`("d"y)/("d"x)` = ex (–A sin x + B cos x) + (A cos x + B sin x) ex
`("d"y)/("d"x)` = ex (–A sin x + B cos x) + y
Again differentiating w.r.t. x, we get
`("d"^2y)/("d"x^2) = "e"^x (-"A" cosx - "B" sinx) + (-"A" sinx + "B"cosx) . "e"^x + ("d"y)/("d"x)`
`("d"^2y)/("d"x^2) = "e"^x ("A" cos x + "B" sin x) + ("d"y)/("d"x) - y + ("d"y)/("d"x)`
`("d"^2y)/("d"x^2) = - y + y + 2("d"y)/("d"x)`
∴ `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y` = 0
APPEARS IN
RELATED QUESTIONS
Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.
Form the differential equation of the family of circles having centre on y-axis and radius 3 units.
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 of the family of curves represented by y2 = (x − c)3.
Form the differential equation from the following primitive where constants are arbitrary:
y2 = 4ax
Form the differential equation from the following primitive where constants are arbitrary:
xy = a2
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 − 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
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 + (y − b)2 = 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 = 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 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:-
\[x\frac{dy}{dx} + y = x^4\]
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 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.
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 curves y = Ae2x + B.e–2x.
Find the differential equation of the family of lines through the origin.
Find the equation of a curve passing through origin and satisfying the differential equation `(1 + x^2) "dy"/"dx" + 2xy` = 4x2
Find the differential equation of system of concentric circles with centre (1, 2).
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.
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
