Advertisements
Advertisements
प्रश्न
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
उत्तर
We have,
\[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
\[ \Rightarrow \frac{1}{\left( 1 + y^2 \right)}dy = \left( 1 + x^2 \right)dx\]
Integrating both sides, we get
\[\int\frac{1}{\left( 1 + y^2 \right)}dy = \int\left( 1 + x^2 \right)dx\]
\[ \Rightarrow \tan^{- 1} y = x + \frac{x^3}{3} + C\]
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Find the particular solution of the differential equation `(1+x^2)dy/dx=(e^(mtan^-1 x)-y)` , give that y=1 when x=0.
Find the particular solution of the differential equation x (1 + y2) dx – y (1 + x2) dy = 0, given that y = 1 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`
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
Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
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 \[\frac{dy}{dx} + 1 = e^{x + y}\], is
The number of arbitrary constants in the particular solution of a differential equation of third order is
Which of the following differential equations has y = x as one of its particular solution?
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 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.
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
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[\frac{dy}{dx} = \left( x + y \right)^2\]
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]
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:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
Solve the following differential equation:-
y dx + (x − y2) dy = 0
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` 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?
