Advertisements
Advertisements
प्रश्न
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Find the general solution of the differential equation:
y2dx + (x2 – xy + y2) dy = 0
Advertisements
उत्तर
The given equation is y2dx + (x2 – xy + y2) dy = 0
⇒ y2dx = – (x2 – xy + y2)dy
⇒ `"dx"/"dy" = - (x^2 - xy + y^2)/y^2`
Since it is a homogeneous differential equation
∴ Put x = vy
⇒ `"dx"/"dy" = "v" + y * "dv"/"dy"`
So, `"v" + y * "dv"/"dy" = - (("v"^2y^2 - "v"y^2 + y^2)/y^2)`
⇒ `"v" + y * "dv"/"dy" = -(y^2("v"^2 - "v" + 1))/y^2`
⇒ `"v" + y * "dv"/"dy" = (-"v"^2 + "v" - 1)`
⇒ `y * "dv"/"dy" = - "v"^2 + "v" - 1 = "v"`
⇒ `y * "dv"/"dy" = - "v"^2 - 1`
⇒ `"dv"/(("v"^2 + 1)) = - "dy"/y`
Integrating both sides, we get
⇒ `int "dv"/(("v"^2 + 1)) = -int "dy"/y`
⇒ `tan^-1"v" = - log y + "c"`
⇒ `tan^-1(x/y) + log y + "c"`
Hence the required solution is `tan^-1(x/y) + log y + "c"`.
संबंधित प्रश्न
The differential equation of `y=c/x+c^2` is :
(a)`x^4(dy/dx)^2-xdy/dx=y`
(b)`(d^2y)/dx^2+xdy/dx+y=0`
(c)`x^3(dy/dx)^2+xdy/dx=y`
(d)`(d^2y)/dx^2+dy/dx-y=0`
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
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 differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Solve the differential equation `dy/dx -y =e^x`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Write the order of the differential equation associated with the primitive y = C1 + C2 ex + C3 e−2x + C4, where C1, C2, C3, C4 are arbitrary constants.
The number of arbitrary constants in the particular solution of a differential equation of third order is
The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
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 the differential equation of all non-horizontal lines in a plane.
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
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)`.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
Solution of differential equation xdy – ydx = 0 represents : ______.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
The general solution of ex cosy dx – ex siny dy = 0 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
Which of the following differential equations has `y = x` as one of its particular solution?
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
