Advertisements
Advertisements
Question
Find the equation of a curve passing through origin and satisfying the differential equation `(1 + x^2) "dy"/"dx" + 2xy` = 4x2
Advertisements
Solution
Given equation is `(1 + x^2) "dy"/"dx" + 2xy` = 4x2
⇒ `"dy"/"dx" + (2x)/(1 + x^2) * y = (4x^2)/(1 + x^2)`
Here, P = `(2x)/(1 + x^2)` and Q = `(4x^2)/(1 + x^2)`
Integrating factor I.F. = `"e"^(int Pdx)`
= `"e"^(int (2x)/(1 + x^2) dx)`
= `"e"^(log(1 + x^2)`
= 1 + x2
∴ Solution is `y xx "I"."F". = int "Q" xx "I"."F". "d"x + "c"`
⇒ `y(1 + x^2) = int (4x^2)/(1 + x^2) xx (1 + x^2) "d"x + "c"`
⇒ `y(1 + x^2) = int 4x^2 "d"x + "c"`
⇒ `y(1 + x^2) = 4/3 x^3 + "c"` ......(i)
Since the curve is passing through origin i.e., (0, 0)
∴ Put y = 0 and x = 0 in equation (i)
0(1 + 0) = `4/3(0)^3 + "c"`
⇒ C = 0
∴ Equation is `y(1 + x^2) = 4/3 x^3`
⇒ y = `(4x^3)/(3(1 + x^2))`
Hence, the required solution is y = `(4x^3)/(3(1 + x^2))`.
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 parabolas having vertex at origin and axis along positive y-axis.
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`
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 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 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 = a (b − x2) by eliminating a and b.
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):
y = eax
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
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 \[\frac{dy}{dx} = e^x \sin x\]
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:-
\[\frac{dy}{dx} - \frac{2xy}{1 + x^2} = x^2 + 2\]
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:-
\[x \log x\frac{dy}{dx} + y = 2 \log x\]
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 of the family of hyperbolas having foci on x-axis and centre at the origin.
Form the differential equation representing the family of curves `y2 = m(a2 - x2) by eliminating the arbitrary constants 'm' and 'a'.
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.
The differential equation representing the family of curves y = A sinx + B cosx is ______.
Find the differential equation of system of concentric circles with centre (1, 2).
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.
Family y = Ax + A3 of curves is represented by the differential equation of degree ______.
Family y = Ax + A3 of curves will correspond to a differential equation of order ______.
The differential equation of the family of curves y2 = 4a(x + a) is ______.
Find the equation of the curve at every point of which the tangent line has a slope of 2x:
Form the differential equation of family of circles having centre on y-axis and raduis 3 units
