Advertisements
Advertisements
प्रश्न
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
पर्याय
k = 0
k > 0
k < 0
none of these
Advertisements
उत्तर
k < 0
We have,
\[ \Rightarrow \frac{dy}{dx} - ky = 0\]
\[ \Rightarrow \frac{dy}{dx} = ky\]
\[ \Rightarrow \frac{1}{y}dy = k dx\]
Integrating both sides, we get
\[\int\frac{1}{y}dy = k\int dx\]
\[ \Rightarrow \log\left| y \right| = kx + C . . . . . \left( 1 \right)\]
Now,
\[y\left( 0 \right) = 1\]
\[ \therefore C = 0\]
\[\text{Putting }C = 0\text{ in }\left( 1 \right),\text{ we get }\]
\[\log\left| y \right| = kx\]
\[ \Rightarrow e^{kx} = y\]
According to the question,
\[ e^{k \infty} = 0\]
\[\text{ Since }e^{- \infty} = 0\]
\[ \therefore k < 0.\]
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
The differential equation of the family of curves y=c1ex+c2e-x is......
(a)`(d^2y)/dx^2+y=0`
(b)`(d^2y)/dx^2-y=0`
(c)`(d^2y)/dx^2+1=0`
(d)`(d^2y)/dx^2-1=0`
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Find the differential equation representing the curve y = cx + c2.
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
Find the particular solution of the differential equation
`tan x * (dy)/(dx) = 2x tan x + x^2 - y`; `(tan x != 0)` given that y = 0 when `x = pi/2`
How many arbitrary constants are there in the general solution of the differential equation of order 3.
Which of the following differential equations has y = x as one of its particular solution?
(1 + y + x2 y) dx + (x + x3) dy = 0
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[\frac{dy}{dx} + y = 4x\]
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
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 the general solution:- \[\frac{dy}{dx} + y = 1\]
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:-
\[\frac{dy}{dx} - y = \cos x\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
Solve the following differential equation:-
\[\left( x + y \right)\frac{dy}{dx} = 1\]
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
If y(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
y = aemx+ be–mx satisfies which of the following differential equation?
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The solution of differential equation coty dx = xdy is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
