Advertisements
Advertisements
प्रश्न
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
Solve the differential equation `2(y + 3) - xy (dy)/(dx)` = 0, given y(1) = – 2.
Advertisements
उत्तर
Given differential equation is `2(y + 3) - xy "dy"/"dx"` = 0
⇒ `xy (dy)/(dx)` = 2y + 6
⇒ `(y/(2y + 6))(dy) = (dx)/x`
⇒ `1/2 (y/(y + 3))(d)y = (dx)/x`
Integrating both sides, we get
⇒ `1/2 int y/(y + 3) dy = int dx/x`
⇒ `1/2 int (y - 3 - 3)/(y + 3) dy = int dx/x`
⇒ `1/2 int (1 - 3/(y + 3)) dy = int (dx)/x`
⇒ `1/2 int dy - 3/2 int 1/(y + 3) dy = int (dx)/x`
⇒ `1/2 y - 3/2 log |y + 3| = log x + c`
Put x = 1, y = –2
⇒ `1/2 (-2) - 3/2 log|-2 + 3| = log(1) + c`
⇒ `-1 - 3/2 log(1) = log(1) + c`
⇒ – 1 – 0 = 0 + c ....[∵ log (1) = 0]
∴ c = –1
∴ Equation is `1/2 y - 3/2 log|y + 3| = log x - 1`
⇒ `y - 3 log |y + 3| = 2 log x - 2`
⇒ `y - 3 log|(y + 3)^3| = log x^2 - 2`
⇒ `log|(y + 3)^3| + log x^2 = y + 2`
⇒ `log|x^2 (y + 3)^3| = y + 2`
⇒ `x^2(y + 3)^3 = e^(y + 2)`
Hence, the required solution is `x^2(y + 3)^3 = e^(y + 2)`
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
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=1 + x + y + xy, given that y = 0 when x = 1.
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:
y = ex + 1 : y″ – y′ = 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 – cos y = x : (y sin y + cos y + x) y′ = y
Show that the general solution of the differential equation `dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0` is given by (x + y + 1) = A (1 - x - y - 2xy), where A is parameter.
Solve the differential equation `cos^2 x dy/dx` + y = tan x
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The number of arbitrary constants in the particular solution of a differential equation of third order is
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
(x2 + 1) dy + (2y − 1) dx = 0
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[\frac{dy}{dx} + y = 4x\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
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:- \[\frac{dy}{dx} + y = 1\]
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:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
Solve the differential equation: `(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = "x"^2 + 2`
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
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 : ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
