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
The solution of the differential equation dy/dx = sec x – y tan x is:
(A) y sec x = tan x + c
(B) y sec x + tan x = c
(C) sec x = y tan x + c
(D) sec x + y tan x = c
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
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:
xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`
Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The number of arbitrary constants in the general solution of differential equation of fourth order is
\[\frac{dy}{dx} + 1 = e^{x + y}\]
\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
The value of c in the particular solution given that y(0) = 0 and k = 0.049 is ______.
Find the general solution of the differential equation:
`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`
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 ______.
