Advertisements
Advertisements
प्रश्न
Solve: ydx – xdy = x2ydx.
Advertisements
उत्तर
Given equation is ydx – xdy = x2ydx.
⇒ ydx – x2y dx = xdy
⇒ y(1 – x2)dx = xdy
⇒ `((1 - x^2)/x)"d"x = "dy"/y`
⇒ `(1/x - x)"d"x = "dy"/y`
Integrating both sides we get
`int(1/x - x)"d"x = int "dy"/y`
⇒ `log x - x^2/2` = log y + log c
⇒ `log x - x^2/2` = log yc
⇒ log y – log c = `x^2/2`
⇒ `log x/(y"c") = x^2/2`
⇒ `x/(y"c") = "e"^(x^2/2)`
⇒ `(y"c")/x = "e"^((-x^2)/2`
⇒ yc = `x"e"^((-x^2)/2`
∴ y = `1/"c" * x"e"^((-x^2)/2`
⇒ y = `"k"x"e"^((-x^2)/2` ......`[because "k" = 1/"c"]`
Hence, the required solution is y = `"k"x"e"^((-x^2)/2`.
APPEARS IN
संबंधित प्रश्न
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
(sin x + cos x) dy + (cos x − sin x) dx = 0
xy (y + 1) dy = (x2 + 1) dx
x cos y dy = (xex log x + ex) dx
(y2 + 1) dx − (x2 + 1) dy = 0
Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.
(x + 2y) dx − (2x − y) dy = 0
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?
The slope of the tangent at a point P (x, y) on a curve is \[\frac{- x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.
Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\] and tangent at any point of which makes an angle tan−1 \[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`
For the following differential equation find the particular solution.
`dy/ dx = (4x + y + 1),
when y = 1, x = 0
Solve the following differential equation.
`dy/dx + 2xy = x`
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
Solve the differential equation
`x + y dy/dx` = x2 + y2
