Advertisements
Advertisements
प्रश्न
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
Advertisements
उत्तर
We have,
\[y = \log x...........(1)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = \frac{1}{x}\]
or,
\[x\frac{dy}{dx} = 1\]
It is the given differential equation.
Thus, \[y = \log x\] satisfies the given differential equation.
Hence, it is a solution.
Also, when \[x = 1, y = \log 1 = 0, i.e., y\left( 1 \right) = 0\]
Hence, \[y = \log x\] is the solution to the given initial value problem.
APPEARS IN
संबंधित प्रश्न
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} = y\]
|
y = ax |
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x + y\frac{dy}{dx} = 0\]
|
\[y = \pm \sqrt{a^2 - x^2}\]
|
(y + xy) dx + (x − xy2) dy = 0
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after `t` seconds.
Solve the following initial value problem:-
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]
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 \[x = \frac{\pi}{2}\]
If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
For each of the following differential equations find the particular solution.
`y (1 + logx)dx/dy - x log x = 0`,
when x=e, y = e2.
Solve the following differential equation.
`dy/dx + y = e ^-x`
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
Choose the correct alternative.
The solution of `x dy/dx = y` log y is
`xy dy/dx = x^2 + 2y^2`
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 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`
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solve the differential equation
`x + y dy/dx` = x2 + y2
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
