Advertisements
Advertisements
प्रश्न
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
Advertisements
उत्तर
We have,
\[y \log y\ dx - x\ dy = 0\]
\[ \Rightarrow y \log y dx = x dy\]
\[ \Rightarrow \frac{1}{x}dx = \frac{1}{y \log y}dy\]
\[ \Rightarrow \frac{1}{y \log y}dy = \frac{1}{x}dx\]
Integrating both sides, we get
\[\int\frac{1}{y \log y}dy = \int\frac{1}{x}dx . . . . . \left( 1 \right)\]
Putting log y = t
\[ \Rightarrow \frac{1}{y}dy = dt\]
Therefore (1) becomes
\[\int\frac{1}{t}dt = \int\frac{1}{x}dx\]
\[ \Rightarrow \log \left( t \right) = \log x + \log C\]
\[ \Rightarrow \log \left( \log y \right) = \log x + \log C\]
\[ \Rightarrow \log \left( \log y \right) = \log Cx\]
\[ \Rightarrow \log y = Cx\]
\[ \Rightarrow y = e^{Cx}\]
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Find the particular solution of the differential equation `(1+x^2)dy/dx=(e^(mtan^-1 x)-y)` , give that y=1 when x=0.
Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y″ – y′ = 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)`
The number of arbitrary constants in the particular solution of a differential equation of third order are ______.
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`
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
The number of arbitrary constants in the general solution of differential equation of fourth order is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is
cos (x + y) dy = dx
(x + y − 1) dy = (x + y) dx
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
\[\frac{dy}{dx} + y = 4x\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
Solve the following differential equation:-
\[\frac{dy}{dx} - y = \cos x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
The solution of differential equation coty dx = xdy is ______.
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
