Advertisements
Advertisements
Question
Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0
Advertisements
Solution
We have:
`e^xsqrt(1−y2)dx+y/x dy=0 `
`e^xsqrt(1−y2)dx=-y/x dy..........(1)`
Separating the variables in equation (1), we get:
`xe^xdx=-y/sqrt(1-y^2)dy.........(2)`
Integrating both sides of equation (2), we have:
`int xe^xdx=-inty/sqrt(1-y^2)dy ............(3)`
`Now,intxe^xdx=xe^x-e^x+C_1=e^x(x-1)+C_1.......(4)`
`"Let " I=-inty/sqrt(1-y^2)dy`
putting `1-y^2=t` we get,
`-2ydy=dt`
`-ydy=dt/2`
`I=1/2intdt/sqrtt`
`=1/2xx2t^(1/2)+C_2`
`=t^(1/2)+C_2`
`=(1-y^2)^(1/2)+C2.......(5)`
Putting the values in equation (3), we get
`e^x(x-1)+C_1=(1-y^2)^(1/2)+C_2`
`e^x(x-1)=(1-y^2)^(1/2)+C, "where " C=C_2-C_1.......(6)`
on putting y=1 and x=0 in equation (6) we get C=-1
The particular solution of the given differential equation is `e^x(x-1)=(1-y^2)-1`
APPEARS IN
RELATED QUESTIONS
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
(1 + y + x2 y) dx + (x + x3) dy = 0
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
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:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
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 \[\frac{dy}{dx} = e^x \sin x.\]
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.
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`.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
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 ______.
Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 0.
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
