Advertisements
Advertisements
Question
Form the differential equation from the relation x2 + 4y2 = 4b2
Advertisements
Solution
Given relation is
x2 + 4y2 = 4b2
Differentiating w.r.t. x, we get
`2x + 4.2y ("d"y)/("d"x) = 0`
∴ `x + 4y ("d"y)/("d"x) = 0`, which is the required differential equation.
APPEARS IN
RELATED QUESTIONS
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
(1 − x2) dy + xy dx = xy2 dx
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\] and tangent at any point of which makes an angle tan−1 \[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.
A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
The solution of `dy/dx + x^2/y^2 = 0` is ______
Solve:
(x + y) dy = a2 dx
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
