Advertisements
Advertisements
प्रश्न
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
Advertisements
उत्तर
`(x + 1) dy/dx -1 = 2e^(-y)`
∴ `(x + 1) dy /dx = 2/e^y + 1`
∴ `(x + 1) dy /dx = ((2+e^y))/e^y `
∴ `e^y /(2+e^y) dy= dx/(1+x)`
Integrating on both sides, we get
`int e^y/(2+e^y) dy = intdx/(1+x)`
∴ log| 2 + ey| = log |1 + x| + log |c|
∴ log |2 + ey| = log |c(1 + x)|
∴ 2 + ey = c (1 + x) ...(i)
When y = 0, x = 1, we have
2 + e0 = c (1 + 1)
∴ 2 + 1 = 2c
∴ c = `3/2`
Substituting c = `3/2` in (i), we get
`2 + e^y = 3/ 2 (1 + x)`
∴ 4 + 2ey = 3 + 3x
∴ 3x - 2ey - 1 = 0, which is the required particular solution.
APPEARS IN
संबंधित प्रश्न
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
(ey + 1) cos x dx + ey sin x dy = 0
x2 dy + y (x + y) dx = 0
2xy dx + (x2 + 2y2) dy = 0
Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]
The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
Solve the following differential equation
sec2 x tan y dx + sec2 y tan x dy = 0
Solution: sec2 x tan y dx + sec2 y tan x dy = 0
∴ `(sec^2x)/tanx "d"x + square` = 0
Integrating, we get
`square + int (sec^2y)/tany "d"y` = log c
Each of these integral is of the type
`int ("f'"(x))/("f"(x)) "d"x` = log |f(x)| + log c
∴ the general solution is
`square + log |tan y|` = log c
∴ log |tan x . tan y| = log c
`square`
This is the general solution.
Find the particular solution of the following differential equation
`("d"y)/("d"x)` = e2y cos x, when x = `pi/6`, y = 0.
Solution: The given D.E. is `("d"y)/("d"x)` = e2y cos x
∴ `1/"e"^(2y) "d"y` = cos x dx
Integrating, we get
`int square "d"y` = cos x dx
∴ `("e"^(-2y))/(-2)` = sin x + c1
∴ e–2y = – 2sin x – 2c1
∴ `square` = c, where c = – 2c1
This is general solution.
When x = `pi/6`, y = 0, we have
`"e"^0 + 2sin pi/6` = c
∴ c = `square`
∴ particular solution is `square`
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
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?
