Advertisements
Advertisements
Question
tan–1x + tan–1y = c is the general solution of the differential equation ______.
Options
`("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
`("d"y)/("d"x) = (1 + x^2)/(1 + y^2)`
(1 + x2)dy + (1 + y2)dx = 0
(1 + x2)dx + (1 + y2)dy = 0
Advertisements
Solution
tan–1x + tan–1y = c is the general solution of the differential equation (1 + x2)dy + (1 + y2)dx = 0.
Explanation:
Given equation is tan–1x + tan–1y = c
Differentiating w.r.t. x, we have
`1/(1 + x^2) + 1/(1 + y^2) * ("d"y)/("d"x)` = 0
⇒ `(1/(1 + y^2)) ("d"y)/("d"x) = -(1/(1 + x^2))`
⇒ `("d"y)/("d"x) = -((1 + y^2)/(1 + x^2))`
⇒ (1 + x2)dy = – (1 + y2)dx
⇒ (1 + x2)dy + (1 + y2)dx = 0.
APPEARS IN
RELATED QUESTIONS
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
Solve the differential equation `cos^2 x dy/dx` + y = tan x
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
The solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\], is
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
Solve the differential equation (x2 − yx2) dy + (y2 + x2y2) dx = 0, given that y = 1, when x = 1.
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
(1 + y + x2 y) dx + (x + x3) dy = 0
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[\frac{dy}{dx} + 5y = \cos 4x\]
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
For the following differential equation, find a particular solution satisfying the given condition:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} - y = \cos x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 2y = \sin x\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x2 + 1) dx, x ≠ 0.
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
If y(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.
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 y2dx + (x2 – xy + y2) dy = 0.
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
The general solution of ex cosy dx – ex siny dy = 0 is ______.
The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx 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 ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
The value of c in the particular solution given that y(0) = 0 and k = 0.049 is ______.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
