Advertisements
Advertisements
Question
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)`
Advertisements
Solution
Given that the slope of the tangent to the curve at (x, y) is `("d"y)/("d"x) = (y - 1)/(x^2 + x)`
⇒ `("d"y)/(y - 1) = ("d"x)/(x^2 + x)`
Integrating both sides, we have
`int ("d"y)/(y - 1) = int ("d"x)/(x^2 + x)`
⇒ `int ("d"y)/(y - 1) = int ("d"x)/(x^2 + x + 1/4 - 1/4)` ...[making perfect square]
⇒ `int ("d"y)/(y - 1) = int ("d"x)/((x + 1/2)^2 - (1/2)^2`
⇒ `log|y - 1| = 1/(2 xx 1/2) log|(x + 1/2 - 1/2)/(x + 1/2 + 1/2)|`
⇒ `log|y - 1| = log|x/(x + 1)| + log "c"`
⇒ `log|y - 1| = log|"c"(x/(x + 1))|`
∴ y – 1 = `("c"x)/(x + 1)`
⇒ `(y - 1)(x + 1)` = cx
Since, the line is passing through the point (1, 0), then (0 – 1) (1 + 1) = c(1)
⇒ c = 2
Hence, the required solution is (y – 1)(x + 1) = 2x.
APPEARS IN
RELATED QUESTIONS
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.
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`
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
Form the differential equation from the following primitive where constants are arbitrary:
xy = a2
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, 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
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):
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):
y = eax
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:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x}\]
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:-
\[\frac{dy}{dx} + y \cos x = e^{\sin x} \cos x\]
Find one-parameter families of solution curves of the following differential equation:-
\[\frac{dy}{dx} \cos^2 x = \tan x - y\]
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:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Write the differential equation representing family of curves y = mx, where m is arbitrary constant.
Write the order of the differential equation representing the family of curves y = ax + a3.
Form the differential equation representing the family of curves y = mx, where m is an arbitrary constant.
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 equation of a curve passing through origin and satisfying the differential equation `(1 + x^2) "dy"/"dx" + 2xy` = 4x2
Form the differential equation by eliminating A and B in Ax2 + By2 = 1
Find the differential equation of system of concentric circles with centre (1, 2).
Find the equation of a curve passing through (2, 1) if the slope of the tangent to the curve at any point (x, y) is `(x^2 + y^2)/(2xy)`.
Family y = Ax + A3 of curves is represented by the differential equation of degree ______.
Form the differential equation of family of circles having centre on y-axis and raduis 3 units
