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
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (x ≠ 0)
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x sin x : xy' = `y + x sqrt (x^2 - y^2)` (x ≠ 0 and x > y or x < -y)
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
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)`
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
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
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
\[\frac{dy}{dx} + 1 = e^{x + y}\]
\[\frac{dy}{dx} = \left( x + y \right)^2\]
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
(x2 + 1) dy + (2y − 1) dx = 0
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
x2 dy + (x2 − xy + y2) dx = 0
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
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:-
\[\left( x + y \right)\frac{dy}{dx} = 1\]
Solve the following differential equation:-
y dx + (x − y2) dy = 0
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The solution of differential equation coty dx = xdy is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Which of the following differential equations has `y = x` as one of its particular solution?
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.
