Advertisements
Advertisements
Question
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Advertisements
Solution
Given:
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Let tan−1y=t
⇒y=tant
`=>dy/dx=sec^2tdt/dx`
Therefore, the equation becomes
(1+tan2t)+(x−et)sec2t `dt/dx=0`
`=>sec^2t+(x-e^t)(sec^2t)dt/dx=0`
`=>1+(x-e^t)dt/dx=0`
`=>(x-e^t)dt/dx=-1`
`=>x-e^t=dx/dt`
`=>dx/dt+1.x=e^t`
If =e∫1.dt
= et
`:. e^t.(dx/dt+1.x)=e^t.e^t`
`=>d/dt(xe^t)=e^(2t)`
Integrating both the sides, we get
`xe^t=inte^(2t)dt`
`=>xe^t=1/2e^(2t)+C " ....(1)"`
Substituting the value of t in (1), we get
`xe^(tan^(1))y=1/2e^(2tan^(-1)y)+C_1`
`=>e^2tan^(-1y)=2xe^(tan^1y)+C`
It is the required general solution.
APPEARS IN
RELATED QUESTIONS
Find the differential equation representing the curve y = cx + c2.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
x + y = tan–1y : y2 y′ + y2 + 1 = 0
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
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 \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
Find the particular solution of the differential equation `(1+y^2)+(x-e^(tan-1 )y)dy/dx=` given that y = 0 when x = 1.
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
\[\frac{dy}{dx} + 1 = e^{x + y}\]
\[\frac{dy}{dx} = \left( x + y \right)^2\]
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
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:- \[x \cos\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 2y = \sin x\]
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 (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`
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)`.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
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.
