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 the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , 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:
y = ex + 1 : y″ – y′ = 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 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 = x sin x : xy' = `y + x sqrt (x^2 - y^2)` (x ≠ 0 and x > y or x < -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 a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
If y = etan x+ (log x)tan x then find dy/dx
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.
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
The solution of the differential equation (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
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:- \[\frac{dy}{dx} = \sin^{- 1} x\]
For the following differential equation, find a particular solution satisfying the given condition:
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]
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:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[\frac{dy}{dx} + 2y = \sin x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
The general solution of ex cosy dx – ex siny dy = 0 is ______.
The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
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 ______.
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
