Advertisements
Advertisements
प्रश्न
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Advertisements
उत्तर
\[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos \left( \frac{y}{x} \right) + x\]
\[ \Rightarrow \frac{dy}{dx} = \frac{y \cos \left( \frac{y}{x} \right) + x}{x \cos \left( \frac{y}{x} \right)}\]
\[\text { This is a homogeneous differential equation } . \]
\[\text { Putting }y = vx and \frac{dy}{dx} = v + x\frac{dv}{dx}, \text { we get }\]
\[v + x\frac{dv}{dx} = \frac{vx \cos v + x}{x \cos v}\]
\[\Rightarrow v + x\frac{dv}{dx} = \frac{v \cos v + 1}{\cos v}\]
\[ \Rightarrow x\frac{dv}{dx} = \frac{v \cos v + 1 - v \cos v}{\cos v}\]
\[ \Rightarrow x\frac{dv}{dx} = \frac{1}{\cos v}\]
\[ \Rightarrow \cos v dv = \frac{1}{x}dx\]
\[\text { Integrating both sides, we get }\]
\[\int\cos v \ dv = \int\frac{1}{x}dx\]
\[ \Rightarrow \sin v = \log \left| x \right| + \log\left| C \right|\]
\[\text { Putting v }= \frac{y}{x}, we get\]
\[\sin\frac{y}{x} = \log \left| Cx \right|\]
\[\text { which is the general solution of the given differential equation } .\]
APPEARS IN
संबंधित प्रश्न
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 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 general solution of a differential equation of fourth order are ______.
If y = etan x+ (log x)tan x then find dy/dx
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`
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.
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\], is
The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, 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
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.
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`.
Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
The general solution of ex cosy dx – ex siny dy = 0 is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
The solution of differential equation coty dx = xdy is ______.
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
