Advertisements
Advertisements
प्रश्न
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
उत्तर
`y = sin^(-1) (6x sqrt(1-9x^2)), -1/(3sqrt2) < x < 1/(3sqrt2)`
`=> y = sin^(-1) (2 xx 3x xx sqrt(1-(3x)^2))`
Putting 3x = sinθ, we have
`y = sin^(-1) (2sin theta sqrt(1- sin^2 theta))`
⇒ y = sin−1 (2sinθ cosθ)
⇒ y = sin−1(sin2θ)
⇒ y = 2θ
`[-1/(3sqrt2) < (sin theta)/3 < 1/(3sqrt2) => - 1/sqrt2 < sin theta < 1/sqrt2 => - pi/4 < theta < pi/4 => -pi/4 < 2theta < pi/2]`
`:. y = 2sin^(-1) 3x`
Differentiating both sides w.r.t x, we get
`(dy)/(dx) = 2 xx 1/(sqrt(1-(3x)^2)) xx 3`
`=> (dy)/dx = 6/sqrt(1- 9x^2)`
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.
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y – cos y = x : (y sin y + cos y + x) y′ = y
Show that the general solution of the differential equation `dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0` is given by (x + y + 1) = A (1 - x - y - 2xy), where A is parameter.
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.`
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
(1 + y + x2 y) dx + (x + x3) dy = 0
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[\frac{dy}{dx} + y = 4x\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
Solve the following differential equation:-
y dx + (x − y2) dy = 0
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
Find the general solution of `"dy"/"dx" + "a"y` = emx
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
Solution of differential equation xdy – ydx = 0 represents : ______.
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
The value of c in the particular solution given that y(0) = 0 and k = 0.049 is ______.
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
