Advertisements
Advertisements
Question
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
Solution
`("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
RELATED QUESTIONS
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = cos x + C : y′ + sin x = 0
Show that the general solution of the differential equation `dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0` is given by (x + y + 1) = A (1 - x - y - 2xy), where A is parameter.
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The general solution of the differential equation \[\frac{dy}{dx} + y\] g' (x) = g (x) g' (x), where g (x) is a given function of x, is
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
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
\[\frac{dy}{dx} + 1 = e^{x + y}\]
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
\[\frac{dy}{dx} - y \tan x = e^x\]
(x2 + 1) dy + (2y − 1) dx = 0
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\frac{dy}{dx} + y = 4x\]
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:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
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:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` 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 a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
