Advertisements
Advertisements
Question
Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.
Advertisements
Solution
The equation of the family of curves is v=A/r+B, where A and B are arbitrary constants.
We have
v=Ar+B
Differentiating both sides with respect to r, we get
`(dv)/(dr)=-A/r^2+0`
`⇒r^2(dv)/(dr)=−A`
Again, differentiating both sides with respect to r, we get
`r^2xx(d^2v)/(d^2r)+2rxx(dv)/(dr)=0`
`⇒r(d^2v)/(d^2r)+2(dv)/(dr)=0`
This is the differential equation representing the family of the given curves
APPEARS IN
RELATED QUESTIONS
Write the integrating factor of the following differential equation:
(1+y2) dx−(tan−1 y−x) dy=0
Find the differential equation of the family of lines passing through the origin.
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
`x/a + y/b = 1`
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = ex (a cos x + b sin x)
Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
Form the differential equation having \[y = \left( \sin^{- 1} x \right)^2 + A \cos^{- 1} x + B\], where A and B are arbitrary constants, as its general solution.
The equation of the curve satisfying the differential equation y (x + y3) dx = x (y3 − x) dy and passing through the point (1, 1) is
Verify that xy = a ex + b e−x + x2 is a solution of the differential equation \[x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0.\]
Show that y = C x + 2C2 is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]
Show that the differential equation of all parabolas which have their axes parallel to y-axis is \[\frac{d^3 y}{d x^3} = 0.\]
\[\frac{dy}{dx} = \frac{1}{x^2 + 4x + 5}\]
\[\frac{dy}{dx} + 4x = e^x\]
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]
tan y dx + tan x dy = 0
(1 + x) y dx + (1 + y) x dy = 0
x cos2 y dx = y cos2 x dy
cosec x (log y) dy + x2y dx = 0
(1 − x2) dy + xy dx = xy2 dx
A solution of the differential equation `("dy"/"dx")^2 - x "dy"/"dx" + y` = 0 is ______.
Solve the differential equation:
cosec3 x dy − cosec y dx = 0
The general solution of the differential equation `(dy)/(dx) + x/y` = 0 is
Solution of the equation 3 tan(θ – 15) = tan(θ + 15) is
Find the general solution of differential equation `(dy)/(dx) = (1 - cosx)/(1 + cosx)`
What is the general solution of differential equation `(dy)/(dx) = sqrt(4 - y^2) (-2 < y < 2)`
The general solution of the differential equation y dx – x dy = 0 is ______.
The general solution of the differential equation ydx – xdy = 0; (Given x, y > 0), is of the form
(Where 'c' is an arbitrary positive constant of integration)
