Advertisements
Advertisements
Question
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
Options
y = tan–1x
y – x = k(1 + xy)
x = tan–1y
tan(xy) = k
Solution
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is y – x = k(1 + xy).
Explanation:
The given differential equation is `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)`
⇒ `("d"y)/(1 + y^2) = ("d"x)/(1 + x^2)`
Integrating both sides, we get
`int ("d"y)/(1 + y^2) = int ("d"x)/(1 + x^2)`
⇒ tan–1y = tan–1x + c
⇒ tan–1y – tan–1x = c
⇒ `tan^-1((y - x)/(1 + xy))` = c
⇒ `(y - x)/(1 + xy)` = tan c
⇒ `((y - x)/(1 + xy))` = k ....[k = tan c]
⇒ y – x = k(1 + xy)
APPEARS IN
RELATED QUESTIONS
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y = sqrt(a^2 - x^2 ) x in (-a,a) : x + y dy/dx = 0(y != 0)`
If y = etan x+ (log x)tan x then find dy/dx
Solve the differential equation `cos^2 x dy/dx` + y = tan x
The general solution of the differential equation \[\frac{dy}{dx} + y\] g' (x) = g (x) g' (x), where g (x) is a given function of x, is
Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] is
The solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\], is
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
\[\frac{dy}{dx} = \left( x + y \right)^2\]
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[\frac{dy}{dx} + 5y = \cos 4x\]
`x cos x(dy)/(dx)+y(x sin x + cos x)=1`
For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[\frac{dy}{dx} - y = \cos x\]
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Solution of differential equation xdy – ydx = 0 represents : ______.
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
