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
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
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 particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Solve the differential equation `dy/dx -y =e^x`
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:
xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
If m and n are the order and degree of the differential equation \[\left( y_2 \right)^5 + \frac{4 \left( y_2 \right)^3}{y_3} + y_3 = x^2 - 1\], then
The number of arbitrary constants in the general solution of differential equation of fourth order is
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] 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.
\[\frac{dy}{dx} = \left( x + y \right)^2\]
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
(1 + y + x2 y) dx + (x + x3) dy = 0
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
(x3 − 2y3) dx + 3x2 y dy = 0
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
Solve the following differential equation:-
y dx + (x − y2) dy = 0
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 solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.
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.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Solution of differential equation xdy – ydx = 0 represents : ______.
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 ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Which of the following differential equations has `y = x` as one of its particular solution?
