Advertisements
Advertisements
प्रश्न
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
Advertisements
उत्तर
`("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1`
⇒ `(d"y")/(2e^-"y" - 1) = (d"x")/("x" + 1)`
⇒ `(e^"y" d"y")/(2 -e^"y") = (d"x")/("x" + 1)`
Integrating both sides, we get:
`int_ (e^"y" d"y")/(2 -e^"y") = log |"x" + 1| + log "C"` .....(1)
Let `2 -e^"y" = t.`
∴ `(d)/(d"y") (2 - e^"y") = (dt)/(d"y")`
⇒ `-e^"y" = (dt)/(d"y")`
⇒ `e^"y" dt = -dt`
Substituting ths value in equation (1), we get:
`int_ (-dt)/(t) = log|"x" + 1| + log "C" `
⇒ `-log|"r"| = log| "C" ("x" + 1)`
⇒ `-log|2 - e^"y"| = log |"C"("x" + 1)|`
⇒ `(1)/(2 - e^"y") = "C" ("x" + 1)`
⇒ `2 - e^"y" = (1)/("C"("x" + 1)` ....(2)
Now, at x = 0 and y = 0, equation (2) becomes:
⇒ `2 - 1 = (1)/("C")`
⇒ `"C" = 1`
Substituting C = 1 in equation (2), we get:
`2 -e^"y" = (1)/("x" + 1)`
⇒ `e^"y" = 2 -(1)/("x" + 1)`
⇒ `e^"y" = (2"x" + 2 - 1)/("x" + 1)`
⇒ `e^"y" = (2"x" + 1)/("x" +1)`
⇒ `"y" log|(2"x" + 1)/("x" + 1)|. ("x" ≠ - 1) `
This is the required particular solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = 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
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Find the differential equation representing the curve y = cx + c2.
Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
\[\frac{dy}{dx} + 2y = \sin 3x\]
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:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
Solve the following differential equation:- \[x \cos\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
Solve the following differential equation:-
\[\left( x + y \right)\frac{dy}{dx} = 1\]
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
If the solution curve of the differential equation `(dy)/(dx) = (x + y - 2)/(x - y)` passes through the point (2, 1) and (k + 1, 2), k > 0, then ______.
Solve the differential equation:
`(xdy - ydx) ysin(y/x) = (ydx + xdy) xcos(y/x)`.
Find the particular solution satisfying the condition that y = π when x = 1.
