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 circles touching the y-axis at the origin.
Form the differential equation of the family of circles having centre on y-axis and radius 3 units.
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`
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.
Form the differential equation of the family of curves represented by y2 = (x − c)3.
Form the differential equation corresponding to y = emx by eliminating m.
Form the differential equation from the following primitive where constants are arbitrary:
y = cx + 2c2 + c3
Form the differential equation from the following primitive where constants are arbitrary:
xy = a2
Form the differential equation corresponding to y2 = a (b − x2) by eliminating a and b.
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):
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):
y = eax
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
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:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x}\]
Find one-parameter families of solution curves of the following differential equation:-
\[x\frac{dy}{dx} + y = x^4\]
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:-
\[\frac{dy}{dx} - \frac{2xy}{1 + x^2} = x^2 + 2\]
Find one-parameter families of solution curves of the following differential equation:-
\[\frac{dy}{dx} + y \cos x = e^{\sin x} \cos x\]
Find one-parameter families of solution curves of the following differential equation:-
\[\frac{dy}{dx} \cos^2 x = \tan x - y\]
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.
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.
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)`
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.
Family y = Ax + A3 of curves will correspond to a differential equation of order ______.
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 ______.
Form the differential equation of family of circles having centre on y-axis and raduis 3 units
