Advertisements
Advertisements
प्रश्न
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Advertisements
उत्तर
y = log x + c
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = 1/x`
∴ `x ("d"y)/("d"x)` = 1
Again, differentiating w.r.t. x, we get
`x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
∴ Given function is a solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
x2 dy + y (x + y) dx = 0
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
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.
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
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.
`dy/dx + y = e ^-x`
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
Choose the correct alternative.
The solution of `x dy/dx = y` log y is
Solve
`dy/dx + 2/ x y = x^2`
Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0
Find the particular solution of the following differential equation
`("d"y)/("d"x)` = e2y cos x, when x = `pi/6`, y = 0.
Solution: The given D.E. is `("d"y)/("d"x)` = e2y cos x
∴ `1/"e"^(2y) "d"y` = cos x dx
Integrating, we get
`int square "d"y` = cos x dx
∴ `("e"^(-2y))/(-2)` = sin x + c1
∴ e–2y = – 2sin x – 2c1
∴ `square` = c, where c = – 2c1
This is general solution.
When x = `pi/6`, y = 0, we have
`"e"^0 + 2sin pi/6` = c
∴ c = `square`
∴ particular solution is `square`
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve the differential equation `"dy"/"dx" + 2xy` = y
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
