Advertisements
Advertisements
प्रश्न
Form the differential equation of all circles which pass through the origin and whose centers lie on X-axis.
Advertisements
उत्तर
Let C(h, 0) be the center of the circle which passes through the origin. Then the radius of the circle is h.
∴ equation of the circle is (x - h)2 + (y - 0)2 = h2
∴ x2 - 2hx + h + y2 = h2
∴ x2 + h2 = 2hx ....(1)
Differentiating both sides w.r.t. x, we get
`"2x" + "2y" "dy"/"dx" = "2h"`
Substituting the value of 2h in equation (1), we get
`"x"^2 + "y"^2 = ("2x" + "2y" "dy"/"dx")"x"`
∴ `"x"^2 + "y"^2 = 2"x"^2 + 2"xy" "dy"/"dx"`
∴ `"2xy" "dy"/"dx" + "x"^2 - "y"^2 = 0`
This is the required D.E.
APPEARS IN
संबंधित प्रश्न
For the differential equation, find the general solution:
`dy/dx + 2y = sin x`
For the differential equation, find the general solution:
`x dy/dx + y - x + xy cot x = 0(x != 0)`
For the differential equation, find the general solution:
`(x + 3y^2) dy/dx = y(y > 0)`
Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
The integrating factor of the differential equation.
`(1 - y^2) dx/dy + yx = ay(-1 < y < 1)` is ______.
Solve the differential equation `(tan^(-1) x- y) dx = (1 + x^2) dy`
Solve the differential equation `x dy/dx + y = x cos x + sin x`, given that y = 1 when `x = pi/2`
x dy = (2y + 2x4 + x2) dx
Solve the differential equation \[\left( x + 2 y^2 \right)\frac{dy}{dx} = y\], given that when x = 2, y = 1.
Solve the differential equation \[\left( y + 3 x^2 \right)\frac{dx}{dy} = x\]
Find the particular solution of the differential equation \[\frac{dx}{dy} + x \cot y = 2y + y^2 \cot y, y ≠ 0\] given that x = 0 when \[y = \frac{\pi}{2}\].
Solve the following differential equation:- \[\left( \cot^{- 1} y + x \right) dy = \left( 1 + y^2 \right) dx\]
Solve the differential equation \[\frac{dy}{dx}\] + y cot x = 2 cos x, given that y = 0 when x = \[\frac{\pi}{2}\] .
Find the integerating factor of the differential equation `x(dy)/(dx) - 2y = 2x^2`
Solve the differential equation: (1 +x2 ) dy + 2xy dx = cot x dx
Solve the differential equation: `(1 + x^2) dy/dx + 2xy - 4x^2 = 0,` subject to the initial condition y(0) = 0.
Solve the following differential equation:
`"x" "dy"/"dx" + "2y" = "x"^2 * log "x"`
Solve the following differential equation:
`("x + y") "dy"/"dx" = 1`
Solve the following differential equation:
dr + (2r cotθ + sin2θ)dθ = 0
Solve the following differential equation:
`(1 + "x"^2) "dy"/"dx" + "y" = "e"^(tan^-1 "x")`
Find the equation of the curve which passes through the origin and has the slope x + 3y - 1 at any point (x, y) on it.
If the slope of the tangent to the curve at each of its point is equal to the sum of abscissa and the product of the abscissa and ordinate of the point. Also, the curve passes through the point (0, 1). Find the equation of the curve.
The integrating factor of the differential equation (1 + x2)dt = (tan-1 x - t)dx is ______.
The slope of the tangent to the curves x = 4t3 + 5, y = t2 - 3 at t = 1 is ______
The equation x2 + yx2 + x + y = 0 represents
The integrating factor of the differential equation `x (dy)/(dx) - y = 2x^2` is
If y = y(x) is the solution of the differential equation, `(dy)/(dx) + 2ytanx = sinx, y(π/3)` = 0, then the maximum value of the function y (x) over R is equal to ______.
Let y = f(x) be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If f(x) satisfies xf'(x) = x2 + f(x) – 2, then the area bounded by f(x) with x-axis between ordinates x = 0 and x = 3 is equal to ______.
Let y = y(x) be the solution curve of the differential equation `(dy)/(dx) + ((2x^2 + 11x + 13)/(x^3 + 6x^2 + 11x + 6)) y = ((x + 3))/(x + 1), x > - 1`, which passes through the point (0, 1). Then y(1) is equal to ______.
If sin x is the integrating factor (IF) of the linear differential equation `dy/dx + Py` = Q then P is ______.
Find the general solution of the differential equation:
`(x^2 + 1) dy/dx + 2xy = sqrt(x^2 + 4)`
Solve the differential equation `dy/dx+2xy=x` by completing the following activity.
Solution: `dy/dx+2xy=x` ...(1)
This is the linear differential equation of the form `dy/dx +Py =Q,"where"`
`P=square` and Q = x
∴ `I.F. = e^(intPdx)=square`
The solution of (1) is given by
`y.(I.F.)=intQ(I.F.)dx+c=intsquare dx+c`
∴ `ye^(x^2) = square`
This is the general solution.
If sec x + tan x is the integrating factor of `dy/dx + Py` = Q, then value of P is ______.
The slope of the tangent to the curve x = sin θ and y = cos 2θ at θ = `π/6` is ______.
