Advertisements
Advertisements
Question
Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis
Solution
Given the circles centre on x-axis and the circle is passing through the origin.
Let it be (r, 0) and its radius r
Equation of the circle is
(x – a)2 + (y – b)2 = r2
(x – r)2 + (y – 0)2 = r2
x2 – 2xr + r2 + y2 = r2
x2 – 2xr + y2 = r2 – r2
x2 – 2xr + y2 = 0 ........(1)
Differentiating equation (1) with respect to ‘x’, we get
2x – 2r + 2y `("d"y)/("d"x)` = 0 dx
2x + 2y `("d"y)/("d"x)` = 2r
`x + y ("d"y)/("d"x)` = r
Substituting r value in equation (1), we get
`x^2 - 2x(x + y ("d"y)/("d"x)) + y^2` = 0
`x^2 - 2x^2 - 2xy ("d"y)/("d"x) + y^2` = 0
`- x^2 - 2xy ("d"y)/("d"x) + y^2` = 0
Multiply by '_', we et
`x^2+ 2xy ("d"y)/("d"x) - y^2` = 0
Which is a required differential equation.
APPEARS IN
RELATED QUESTIONS
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
Ax2 + By2 = 1
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = A cos (log x) + B sin (log x)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = e−2x (A cos x + B sin x)
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = xm; `"x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my" = 0`
For the following differential equation find the particular solution satisfying the given condition:
`y(1 + log x) dx/dy - x log x = 0, y = e^2,` when x = e
For the following differential equation find the particular solution satisfying the given condition:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`
Choose the correct option from the given alternatives:
The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
Solve the following differential equation:
`"dy"/"dx" = "x"^2"y" + "y"`
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
Find the particular solution of the following differential equation:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
Select and write the correct alternative from the given option for the question
Solution of the equation `x ("d"y)/("d"x)` = y log y is
Find the differential equation of family of lines making equal intercepts on coordinate axes
Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis
Find the differential equation of the curve represented by xy = aex + be–x + x2
If m and n are respectively the order and degree of the differential equation of the family of parabolas with focus at the origin and X-axis as its axis, then mn - m + n = ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
For the curve C: (x2 + y2 – 3) + (x2 – y2 – 1)5 = 0, the value of 3y' – y3 y", at the point (α, α), α < 0, on C, is equal to ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
