Advertisements
Advertisements
प्रश्न
Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
Advertisements
उत्तर
The given equation is
(1 + e2x)dy + (1 + y2) ex dx = 0
⇒ `dy/(1 + y^2) + e^x/(1 + e^(2x)) dx = 0`
Integrating, `int dy/(1 + y^2) + int e^x/(1 + e^(2x)) dx = C_1`
⇒ `tan^-1 y + I = C_1` ....(1)
Where `I = int e^x/(1 + e^(2x)) dx`
Put ex = t
⇒ ex dx = dt
∴ `I = int dt/(1 + t^2) = tan^-1 t + C_2`
= tan-1 ex + C2
From (1), tan-1 y + tan-1 ex + C2 = C1
or tan-1 y + tan-1 ex = C .....(2)
When x = 0, y = 1,
∴ tan-1 (1) + tan-1 (e0) = C
⇒ tan-1 (1) + tan-1 (1) = C
⇒ `pi/4 + pi/4 = C`
⇒ `C = pi/2`
Putting in (2) `tan^-1 y + tan^-1 e^x = pi/2,`
Which is the required solution.
APPEARS IN
संबंधित प्रश्न
Find the differential equation of the family of lines passing through the origin.
Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
`x/a + y/b = 1`
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = ex (a cos x + b sin x)
Solve the differential equation `ye^(x/y) dx = (xe^(x/y) + y^2)dy, (y != 0)`
Find a particular solution of the differential equation (x - y) (dx + dy) = dx - dy, given that y = -1, when x = 0. (Hint: put x - y = t)
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is ______.
Find the differential equation of all the circles which pass through the origin and whose centres lie on x-axis.
Form the differential equation having \[y = \left( \sin^{- 1} x \right)^2 + A \cos^{- 1} x + B\], where A and B are arbitrary constants, as its general solution.
Show that y = C x + 2C2 is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]
Show that y2 − x2 − xy = a is a solution of the differential equation \[\left( x - 2y \right)\frac{dy}{dx} + 2x + y = 0.\]
Verify that y = A cos x + sin x satisfies the differential equation \[\cos x\frac{dy}{dx} + \left( \sin x \right)y=1.\]
Show that the differential equation of all parabolas which have their axes parallel to y-axis is \[\frac{d^3 y}{d x^3} = 0.\]
From x2 + y2 + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.
\[\frac{dy}{dx} = y^2 + 2y + 2\]
\[\frac{dy}{dx} + 4x = e^x\]
\[\frac{dy}{dx} = x^2 e^x\]
\[(\tan^2 x + 2\tan x + 5)\frac{dy}{dx} = 2(1+\tan x)\sec^2x\]
tan y dx + tan x dy = 0
cosec x (log y) dy + x2y dx = 0
A solution of the differential equation `("dy"/"dx")^2 - x "dy"/"dx" + y` = 0 is ______.
Find the general solution of the following differential equation:
`x (dy)/(dx) = y - xsin(y/x)`
If n is any integer, then the general solution of the equation `cos x - sin x = 1/sqrt(2)` is
Solution of the equation 3 tan(θ – 15) = tan(θ + 15) is
The general solution of the differential equation of the type `(dx)/(dy) + p_1y = theta_1` is
The general solution of the differential equation `(ydx - xdy)/y` = 0
What is the general solution of differential equation `(dy)/(dx) = sqrt(4 - y^2) (-2 < y < 2)`
The general solution of the differential equation y dx – x dy = 0 is ______.
