Advertisements
Advertisements
प्रश्न
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)`
Advertisements
उत्तर
We have, `y = sqrt (a^2 - x^2)` ...(i)
Differentiating (I) w.r.t. x, we get
`y' = (1xx (-2x))/(2sqrt(a^2 - x^2))`
⇒ `y' = (-x)/sqrt (a^2 - x^2)`
⇒ `y' = (-x)/y` (Using (i))
⇒ yy' = -x
⇒ x + yy' = 0
∴ `y = sqrt (a^2 - x^2)` is a solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
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 dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
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 = Ax : xy′ = y (x ≠ 0)
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
The number of arbitrary constants in the particular solution of a differential equation of third order are ______.
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is
The solution of the differential equation (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The number of arbitrary constants in the particular solution of a differential equation of third order is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
cos (x + y) dy = dx
\[\frac{dy}{dx} - y \tan x = e^x\]
(x2 + 1) dy + (2y − 1) dx = 0
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
x2 dy + (x2 − xy + y2) dx = 0
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) 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.`
Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.
Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
