Advertisements
Advertisements
प्रश्न
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
Advertisements
उत्तर
We have,
\[ x^2 - y^2 = C^2 \]
Differentiating with respect to x, we get
\[2x - 2y\frac{dy}{dx} = 0\]
\[ \Rightarrow 2x = 2y\frac{dy}{dx}\]
\[ \Rightarrow x dx = y dy\]
\[ \Rightarrow x dx - y dy = 0\]
Hence, x dx - y dy = 0 is the required differential equation .
APPEARS IN
संबंधित प्रश्न
Show that the differential equation of which \[y = 2\left( x^2 - 1 \right) + c e^{- x^2}\] is a solution is \[\frac{dy}{dx} + 2xy = 4 x^3\]
(sin x + cos x) dy + (cos x − sin x) dx = 0
(ey + 1) cos x dx + ey sin x dy = 0
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
The solution of the differential equation y1 y3 = y22 is
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
y2 dx + (x2 − xy + y2) dy = 0
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
The price of six different commodities for years 2009 and year 2011 are as follows:
| Commodities | A | B | C | D | E | F |
|
Price in 2009 (₹) |
35 | 80 | 25 | 30 | 80 | x |
| Price in 2011 (₹) | 50 | y | 45 | 70 | 120 | 105 |
The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| xy = log y + k | y' (1 - xy) = y2 |
Find the differential equation whose general solution is
x3 + y3 = 35ax.
For each of the following differential equations find the particular solution.
(x − y2 x) dx − (y + x2 y) dy = 0, when x = 2, y = 0
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Solve the following differential equation.
`dy/dx + 2xy = x`
The solution of `dy/dx + x^2/y^2 = 0` is ______
State whether the following is True or False:
The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.
y dx – x dy + log x dx = 0
Select and write the correct alternative from the given option for the question
Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
