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
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
Differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
(y2 + 1) dx − (x2 + 1) dy = 0
3x2 dy = (3xy + y2) dx
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
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?
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).
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
`dy/dx + 2xy = x`
Solve the following differential equation.
`(x + a) dy/dx = – y + a`
Solve: ydx – xdy = x2ydx.
Which of the following defines a differential equation?
