Advertisements
Advertisements
Question
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Advertisements
Solution
Given equation is `(1 + y^2) + (x - "e"^(tan^(-1) y)) "dy"/"dx"` = 0
⇒ `(x - "e"^(tan^-1y)) "dy"/"dx" = -(1 + y^2)`
⇒ `"dy"/"dx" = (-(1 + y^2))/(x - "e"^(tan^-1 y))`
⇒ `"dx"/"dy" = (x - "e"^(tan^-1y))/(-(1 + y^2))`
⇒ `"dx"/"dy" = - x/((1 + y^2)) + ("e"^(tan^-1y))/(1 + y^2)`
⇒ `"dx"/"dy" + x/((1 + y^2)) = ("e"^(tan^-1 y))/(1 + y^2)`
Here, P = `1/(1 + y^2)` and Q = `("e"^(tan^-1 y))/(1 + y^2)`
∴ Integrating factor I.F. = `"e"^(int Pdy)`
= `"e"^(int 1/(1 + y^2) "d"y)`
= `"e"^(tan^-1 y)`
∴ Solution is `x . "I"."F". = int "Q". "I"."F". "d"y + "c"`
⇒ `x . "e"^(tan^-1 y) = int ("e"^(tan^-1 y))/(1 + y^2) * "e"^(tan^-1 y) "dy" + "c"`
Put `"e"^(tan^-1 y)` = t
∴ `"e"^(tan^-1 y) * 1/(1 + y^2) "dy"` = dt
∴ `x . "e"^(tan^-1 y) = int "t" . "dt" + "c"`
⇒ `x . "e"^(tan^-1 y) = 1/2 "t"^2 + "c"`
⇒ `x . "e"^(tan^-1 y) = 1/2 ("e"^(tan^-1 y))^2 + "c"`
⇒ x = `1/2 ("e"^(tan^-1 y)) + "c"/("e"^(tan^-1 y))`
⇒ 2x = `"e"^(tan^-1 y) + (2"c")/("e"^(tan^-1 y)`
⇒ `2x . "e"^(tan^-1 y) = ("e"^(tan^-1y))^2 + 2"c"`
Hence, this is the required general solution.
APPEARS IN
RELATED QUESTIONS
The differential equation of `y=c/x+c^2` is :
(a)`x^4(dy/dx)^2-xdy/dx=y`
(b)`(d^2y)/dx^2+xdy/dx+y=0`
(c)`x^3(dy/dx)^2+xdy/dx=y`
(d)`(d^2y)/dx^2+dy/dx-y=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
Find the particular solution of the differential equation
`tan x * (dy)/(dx) = 2x tan x + x^2 - y`; `(tan x != 0)` given that y = 0 when `x = pi/2`
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 solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\], is
The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
x2 dy + (x2 − xy + y2) dx = 0
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
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:-
\[\frac{dy}{dx} - y = \cos x\]
Solve the following differential equation:-
y dx + (x − y2) dy = 0
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
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.
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.
Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
The solution of differential equation coty dx = xdy is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Find the general solution of the differential equation:
`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
Solve the differential equation:
`(xdy - ydx) ysin(y/x) = (ydx + xdy) xcos(y/x)`.
Find the particular solution satisfying the condition that y = π when x = 1.
