Advertisements
Advertisements
प्रश्न
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
Advertisements
उत्तर
Given equation is `(x + 2y^3) "dy"/"dx"` = y
⇒ `"dy"/"dx" = y/(x + 2y^3)`
⇒ `"dx"/"dy" = (x + 2y^3)/y`
⇒ `"dx"/"dy" = x/y + (2y^3)/y`
⇒ `"dx"/"dy" - x/y` = 2y3
Here P = `- 1/y` and Q = 2y2.
∴ Integrating factor I.F. = `"e"^(intPdy)`
= `"e"^(int 1/y dy)`
= `"e"^(-log y)`
= `"e"^(log 1/y)`
= `1/y`.
So the solution of the equation is
x.I.F. = `int "Q"."I"."F". "d"y + "c"`
`x . 1/y = int 2y^2 . 1/y "d"y + "c"`
⇒ `x/y = 2 int y "d"y + "c"`
⇒ `x/y = 2. y^2/2 + "c"`
⇒ `x/y = y^2 + "c"`
So x = y3 + cy = y(y2 + c)
Hence, the required solution is x = y(y2 + c).
APPEARS IN
संबंधित प्रश्न
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
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 particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = cos x + C : y′ + sin x = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`
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
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] is
The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, 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
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
cos (x + y) dy = dx
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
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:-
(1 + x2) dy + 2xy dx = cot x dx
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 general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.
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.
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 ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
