Advertisements
Advertisements
प्रश्न
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Advertisements
उत्तर
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
संबंधित प्रश्न
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = cos x + C : y′ + sin x = 0
Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
If y = etan x+ (log x)tan x then find dy/dx
Solve the differential equation `cos^2 x dy/dx` + y = tan x
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
x (e2y − 1) dy + (x2 − 1) ey dx = 0
(1 + y + x2 y) dx + (x + x3) dy = 0
x2 dy + (x2 − xy + y2) dx = 0
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
For the following differential equation, find a particular solution satisfying the given condition:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
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 ______.
