Advertisements
Advertisements
प्रश्न
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
Advertisements
उत्तर
The given equation is
`(x + 1) dy/dx = 2e^-y - 1`
⇒ `dy/(2e^-y - 1) = dx/(x + 1)` ....(1)
Integrating, we get `int dy/(2e^-y - 1) = int dx/(x + 1) + C`
⇒ `int dy/(2e^-y - 1) =log |x + 1| + C`
Now, `I = int dy/ (2e^-y - 1) = int e^y/(2 - e^y) dy`
Put ey = t so that ey dy = dt
∴ `I = int dt/(2-t) = - log |2 - t| = - log |2 - e^y|`
From (1), - log |2 - ey|
= log |x + 1| + C ....(2)
When x = 0. y = 0
∴ - log |2 - 1| = log |0 + 1| + C
⇒ - log |1| = log |1| + C
⇒ 0 = 0 + C
⇒ C = 0
Putting in (2), - log |2 ey| = log |x + 1|
⇒ `log |2 - e^y| = log |1/ (x + 1)|`
⇒ `2 e^y = 1/(x + 1)`
⇒ `e^y = 2 - 1/ (x + 1) = (2x + 1)/(x + 1)`
⇒ `y = log |(2x + 1)/ (x + 1)|, x ne -1`
Which is the required solution.
APPEARS IN
संबंधित प्रश्न
The differential equation of the family of curves y=c1ex+c2e-x is......
(a)`(d^2y)/dx^2+y=0`
(b)`(d^2y)/dx^2-y=0`
(c)`(d^2y)/dx^2+1=0`
(d)`(d^2y)/dx^2-1=0`
Find the differential equation representing the curve y = cx + c2.
Find the particular solution of the differential equation x (1 + y2) dx – y (1 + x2) dy = 0, given that y = 1 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y – cos y = x : (y sin y + cos y + x) y′ = y
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
x + y = tan–1y : y2 y′ + y2 + 1 = 0
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is
The number of arbitrary constants in the particular solution of a differential equation of third order is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
\[\frac{dy}{dx} + 1 = e^{x + y}\]
\[\frac{dy}{dx} = \left( x + y \right)^2\]
(x3 − 2y3) dx + 3x2 y dy = 0
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[\frac{dy}{dx} - y = \cos x\]
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Solve the differential equation: `(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = "x"^2 + 2`
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
Find the differential equation of all non-horizontal lines in a plane.
If y(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 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`
