Advertisements
Advertisements
प्रश्न
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
Advertisements
उत्तर
The equation of the parabola having vertex at origin and axis along the positive direction of x-axis is given by
y2 =4ax .....(1)
Since there is only one parameter, so we differentiate it only once.
Differentiating with respect to x, we get
\[2y\frac{dy}{dx} = 4a\]
Substituting the value of 4a in (1), we get
\[y^2 = 2y\frac{dy}{dx} \times x\]
\[ \Rightarrow y^2 = 2xy\frac{dy}{dx}\]
\[ \Rightarrow y^2 - 2xy\frac{dy}{dx} = 0\]
\[\]
APPEARS IN
संबंधित प्रश्न
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
Show that y = AeBx is a solution of the differential equation
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 1\] Function y = sin x + cos x
x cos y dy = (xex log x + ex) dx
x cos2 y dx = y cos2 x dy
xy dy = (y − 1) (x + 1) dx
tan y dx + sec2 y tan x dy = 0
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
3x2 dy = (3xy + y2) dx
Solve the following initial value problem:-
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
Solve the following initial value problem:-
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2 - y; \tan x \neq 0\] given that y = 0 when \[x = \frac{\pi}{2}\]
The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).
Find the equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\] at any point (x, y) on it.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Solve the following differential equation.
`dy/dx = x^2 y + y`
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Solve the following differential equation.
`dy/dx + 2xy = x`
Solve:
(x + y) dy = a2 dx
Solve
`dy/dx + 2/ x y = x^2`
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve the following differential equation y2dx + (xy + x2) dy = 0
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
Which of the following is an example of an ordinary differential equation?
