Advertisements
Advertisements
Question
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Advertisements
Solution
The equation of the family of ellipses having centre at the origin and foci on the x-axis is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.........(1)\]
where a and b are the parameters.
As this equation contains two parameters, we shall get a second-order differential equation.
Differentiating (1) with respect to x, we get
\[\frac{2x}{a^2} + \frac{2y}{b^2}\frac{dy}{dx} = 0..........(2)\]
Differentiating (2) with respect to x, we get
\[\frac{2}{a^2} + \frac{2}{b^2}\left[ \left( \frac{dy}{dx} \right)^2 + y\frac{d^2 y}{d x^2} \right] = 0\]
\[ \Rightarrow \frac{2}{a^2} = - \frac{2}{b^2}\left[ \left( \frac{dy}{dx} \right)^2 + y\frac{d^2 y}{d x^2} \right]\]
\[ \Rightarrow \frac{b^2}{a^2} = - \left[ \left( \frac{dy}{dx} \right)^2 + y\left( \frac{d^2 y}{d x^2} \right) \right] .........(3)\]
Now, from (2), we get
\[\frac{x}{a^2} = - \frac{y}{b^2}\frac{dy}{dx}\]
\[ \Rightarrow \frac{b^2}{a^2} = - \frac{y}{x}\frac{dy}{dx} ..........(4)\]
From (3) and (4), we get
\[- \frac{y}{x}\frac{dy}{dx} = - \left[ \left( \frac{dy}{dx} \right)^2 + y\left( \frac{d^2 y}{d x^2} \right) \right]\]
\[ \Rightarrow \frac{y}{x}\frac{dy}{dx} = \left[ \left( \frac{dy}{dx} \right)^2 + y\left( \frac{d^2 y}{d x^2} \right) \right]\]
\[ \Rightarrow y\frac{dy}{dx} = x \left( \frac{dy}{dx} \right)^2 + xy\left( \frac{d^2 y}{d x^2} \right)\]
\[ \Rightarrow xy\frac{d^2 y}{d x^2} + x \left( \frac{dy}{dx} \right)^2 - y\frac{dy}{dx} = 0\]
It is the required differential equation.
APPEARS IN
RELATED QUESTIONS
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
(1 − x2) dy + xy dx = xy2 dx
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).
A population grows at the rate of 5% per year. How long does it take for the population to double?
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
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
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] 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 representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
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` |
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = ex | `dy/ dx= y` |
Solve the following differential equation.
`dy/dx + y` = 3
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
The solution of `dy/ dx` = 1 is ______.
Choose the correct alternative.
Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in
Solve
`dy/dx + 2/ x y = x^2`
Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
