Advertisements
Advertisements
Question
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Advertisements
Solution
We know that (x−a)2+(y−b)2=r2 represents a circle with centre (a, b) and radius r.
Since the circle lies in the 2nd quadrant, and touches the coordinate axes, thus a < 0, b > 0 and |a| = |b| = r.
So, the equation becomes (x+a)2+(y−a)2=a2 .....(1)
Differentiating this equation w.r.t. x, we get
`2(x+a)+2(y−a)dy/dx=0`
`⇒dy/dx=(−x+a)/(y−a)`
Putting `dy/dx=y',` we get
`y'=(−x+a)/(y−a)`
`⇒yy'−ay'+x+a=0`
`⇒yy'+x=ay'−a`
`⇒a=(x+yy')/(y'−1)`
Substituting this value of a in (1), we get
`(x−(x+yy')/(y'−1))^2+(y−(x+yy')/(y'−1))^2=((x+yy')/(y'−1))^2`
`⇒(xy'−x−x−yy')^2+(yy'−y−x−yy')^2=(x+yy')^2`
`⇒[y'(x−y)−2x]^2+(x+y)^2=(x+yy')^2`
`⇒(y')^2(x^2−2xy+y^2)−4x^2y'+4xyy'+4x^2+x^2+2xy+y^2=x^2+2xyy'+y^2(y')^2`
`⇒(y')^2(x^2−2xy)+2xy'(−2x+y)+4x^2+2xy+y^2=0`
This is the required differential equation of the family of circles in the second quadrant and touching the coordinate axes.
APPEARS IN
RELATED QUESTIONS
The solution of the differential equation dy/dx = sec x – y tan x is:
(A) y sec x = tan x + c
(B) y sec x + tan x = c
(C) sec x = y tan x + c
(D) sec x + y tan x = c
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y″ – y′ = 0
If y = etan x+ (log x)tan x then find dy/dx
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
For the following differential equation, find a particular solution satisfying the given condition:
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
If the solution curve of the differential equation `(dy)/(dx) = (x + y - 2)/(x - y)` passes through the point (2, 1) and (k + 1, 2), k > 0, then ______.
