Advertisements
Advertisements
प्रश्न
Find the differential equation of the family of curves y = Ae2x + Be−2x, where A and B are arbitrary constants.
Advertisements
उत्तर
The equation of the family of curves is \[y = A e^{2x} + B e^{- 2x}\] ...(1)
where \[A\text{ and }B\] are arbitrary constants.
This equation contains two arbitrary constants, so we shall get a differential equation of second order.
Differentiating equation (1) with respect to x, we get
Differentiating equation (2) with respect to x, we get
\[\frac{d^2 y}{d x^2} = 4A e^{2x} + 4B e^{- 2x} \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 4\left( A e^{2x} + B e^{- 2x} \right)\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 4y\]
It is the required differential equation .
APPEARS IN
संबंधित प्रश्न
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.
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 corresponding to y = emx by eliminating m.
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 = 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):
(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 = 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 \[\frac{dy}{dx} = e^x \sin x\]
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:-
\[x\frac{dy}{dx} + y = x^4\]
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:-
\[e^{- y} \sec^2 y dy = dx + x dy\]
Find one-parameter families of solution curves of the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
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 `y2 = m(a2 - x2) by eliminating the arbitrary constants 'm' and 'a'.
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.
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 ______.
The differential equation representing the family of curves y = A sinx + B cosx is ______.
Form the differential equation of all circles which pass through origin and whose centres lie on y-axis.
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 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 of the family of curves x2 + y2 – 2ay = 0, where a is arbitrary constant, is ______.
Family y = Ax + A3 of curves will correspond to a differential equation of order ______.
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:
From the differential equation of the family of circles touching the y-axis at origin
