Advertisements
Advertisements
Question
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + y2 = ax3
Advertisements
Solution
The equation of family of curves is \[x^2 + y^2 = a x^3........(1)\]
where `a` is a parameter.
As this equation has only one arbitrary constant, we shall get a differential equation of first order.
Differentiating (1) with respect to x, we get
\[2x + 2y\frac{dy}{dx} = 3a x^2 \]
\[ \Rightarrow 2x + 2y\frac{dy}{dx} = 3\left( \frac{x^2 + y^2}{x^3} \right) x^2 ........\left[\text{Using}\left( 1 \right) \right]\]
\[ \Rightarrow 2x + 2y\frac{dy}{dx} = 3\frac{x^2 + y^2}{x}\]
\[ \Rightarrow 2 x^2 + 2xy\frac{dy}{dx} = 3 x^2 + 3 y^2 \]
\[ \Rightarrow 2xy\frac{dy}{dx} = x^2 + 3 y^2 \]
It is the required differential equation.
APPEARS IN
RELATED QUESTIONS
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Form the differential equation representing the family of curves given by (x – a)2 + 2y2 = a2, where a is an arbitrary constant.
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
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 = cx + 2c2 + c3
Form the differential equation from the following primitive where constants are arbitrary:
y = ax2 + bx + c
Form the differential equation corresponding to (x − a)2 + (y − b)2 = r2 by eliminating a and b.
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):
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y2 = 4a (x − b)
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:-
\[\frac{dy}{dx} + 3y = e^{mx}\], m is a given real number.
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:-
\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]
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:-
\[e^{- y} \sec^2 y dy = dx + x dy\]
Find one-parameter families of solution curves of the following differential equation:-
\[x \log x\frac{dy}{dx} + y = 2 \log x\]
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 `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.
Find the differential equation of the family of lines through the origin.
Find the equation of a curve whose tangent at any point on it, different from origin, has slope `y + y/x`.
Find the differential equation of system of concentric circles with centre (1, 2).
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 is represented by the differential equation of degree ______.
The differential equation `y ("d"y)/("d"x) + "c"` represents: ______.
The differential equation of the family of curves x2 + y2 – 2ay = 0, where a is arbitrary constant, is ______.
The differential equation of the family of curves y2 = 4a(x + a) is ______.
The area above the x-axis and under the curve `y = sqrt(1/x - 1)` for `1/2 ≤ x ≤ 1` is:
Form the differential equation of family of circles having centre on y-axis and raduis 3 units
