Advertisements
Advertisements
Question
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
Advertisements
Solution
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
RELATED QUESTIONS
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
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
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y″ – y′ = 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:
x + y = tan–1y : y2 y′ + y2 + 1 = 0
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
Write the order of the differential equation associated with the primitive y = C1 + C2 ex + C3 e−2x + C4, where C1, C2, C3, C4 are arbitrary constants.
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] is
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
Find the particular solution of the differential equation `(1+y^2)+(x-e^(tan-1 )y)dy/dx=` given that y = 0 when x = 1.
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
cos (x + y) dy = dx
\[\frac{dy}{dx} - y \tan x = e^x\]
(x2 + 1) dy + (2y − 1) dx = 0
(x3 − 2y3) dx + 3x2 y dy = 0
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
For the following differential equation, find a particular solution satisfying the given condition:
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
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`.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 0.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
