Advertisements
Advertisements
Question
Show that the differential equation of all parabolas which have their axes parallel to y-axis is \[\frac{d^3 y}{d x^3} = 0.\]
Solution
The equation of the family of parabolas axis parallel to y-axis is given by
\[\left( x - \beta \right)^2 = 4a\left( y - \alpha \right)............(1)\]
Here,
\[\alpha\text{ and }\beta\] are two arbitrary constants.
Differentiating (1) with respect to x, we get
\[2\left( x - \beta \right) = 4a\frac{dy}{dx}\]
\[ \Rightarrow 1 = 2a\frac{d^2 y}{d x^2}\]
\[ \Rightarrow 0 = 2a\frac{d^3 y}{d x^3}\]
\[ \Rightarrow \frac{d^3 y}{d x^3} = 0\]
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.
y2 = a (b2 – x2)
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = a e3x + b e– 2x
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = e2x (a + bx)
Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
Solve the differential equation `ye^(x/y) dx = (xe^(x/y) + y^2)dy, (y != 0)`
The general solution of a differential equation of the type `dx/dy + P_1 x = Q_1` is ______.
Find the differential equation of all the circles which pass through the origin and whose centres lie on y-axis.
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 y2 − x2 − xy = a is a solution of the differential equation \[\left( x - 2y \right)\frac{dy}{dx} + 2x + y = 0.\]
Verify that y = A cos x + sin x satisfies the differential equation \[\cos x\frac{dy}{dx} + \left( \sin x \right)y=1.\]
Find the differential equation corresponding to y = ae2x + be−3x + cex where a, b, c are arbitrary constants.
From x2 + y2 + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.
\[\frac{dy}{dx} = \sin^3 x \cos^4 x + x\sqrt{x + 1}\]
\[\frac{dy}{dx} = \frac{1}{x^2 + 4x + 5}\]
\[\frac{dy}{dx} = x^2 e^x\]
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]
\[(\tan^2 x + 2\tan x + 5)\frac{dy}{dx} = 2(1+\tan x)\sec^2x\]
\[\frac{dy}{dx} = \sin^3 x \cos^2 x + x e^x\]
tan y dx + tan x dy = 0
x cos2 y dx = y cos2 x dy
cosec x (log y) dy + x2y dx = 0
Solution of the equation 3 tan(θ – 15) = tan(θ + 15) is
The number of arbitrary constant in the general solution of a differential equation of fourth order are
The general solution of the differential equation `(dy)/(dx) = e^(x + y)` is
The general solution of the differential equation `(ydx - xdy)/y` = 0
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)
