Advertisements
Advertisements
Question
Find the differential equation of the family of curves y = Ae2x + B.e–2x.
Advertisements
Solution
y = Ae2x + B.e–2x.
`("d"y)/("d"x) = 2"Ae"^(2x) - 2"B"e"^(-2x)` and `("d"^2y)/("dx"^2) = 4"Ae"^(2x) + 4"Be"^(-2x)`
Thus `("d"^2y)/("dx"^2) = 4y`
i.e., `("d"^2y)/("dx"^2) - 4y` = 0.
APPEARS IN
RELATED QUESTIONS
Form the differential equation of the family of circles touching the y-axis at the origin.
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 of the family of circles in the first quadrant which touch the coordinate axes.
For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3
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 of the family of curves represented by y2 = (x − c)3.
Form the differential equation from the following primitive where constants are arbitrary:
y = ax2 + bx + c
Find the differential equation of the family of curves y = Ae2x + Be−2x, where A and B are arbitrary constants.
Find the differential equation of the family of curves, x = A cos nt + B sin nt, 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
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):
y2 = 4ax
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):
(x − a)2 − y2 = 1
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + y2 = ax3
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 \[\frac{dy}{dx} = e^x \sin x\]
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:-
\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]
Find one-parameter families of solution curves of the following differential equation:-
\[x \log x\frac{dy}{dx} + y = 2 \log x\]
The differential equation which represents the family of curves y = eCx is
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 differential equation of the family of lines through the origin.
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.
Form the differential equation of all circles which pass through origin and whose centres lie on y-axis.
Form the differential equation by eliminating A and B in Ax2 + By2 = 1
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.
Family y = Ax + A3 of curves is represented by the differential equation of degree ______.
The differential equation representing the family of circles x2 + (y – a)2 = a2 will be of order two.
From the differential equation of the family of circles touching the y-axis at origin
Form the differential equation of family of circles having centre on y-axis and raduis 3 units
Form the differential equation of the family of hyperbola having foci on x-axis and centre at origin.
