Advertisements
Advertisements
प्रश्न
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
Advertisements
उत्तर
We have
\[\frac{dy}{dx} = 2^{- y} \]
\[ \Rightarrow \frac{dy}{2^{- y}} = dx\]
\[ \Rightarrow 2^y dy = dx\]
Integrating both sides, we get
\[\int 2^y dy = \int dx\]
\[ \Rightarrow \frac{2^y}{\log2} = x + c\]
\[ \Rightarrow 2^y = x\log2 + k, \text { where } k = c\log2\]
APPEARS IN
संबंधित प्रश्न
If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`
Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
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
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y = sqrt(a^2 - x^2 ) x in (-a,a) : x + y dy/dx = 0(y != 0)`
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
(x + y − 1) dy = (x + y) dx
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
\[\frac{dy}{dx} + 5y = \cos 4x\]
`x cos x(dy)/(dx)+y(x sin x + cos x)=1`
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:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x2 + 1) dx, x ≠ 0.
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
The general solution of ex cosy dx – ex siny dy = 0 is ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The solution of differential equation coty dx = xdy is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Find the general solution of the differential equation:
`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
