Advertisements
Advertisements
Question
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
Advertisements
Solution
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
RELATED QUESTIONS
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Verify that y = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
2xy dx + (x2 + 2y2) dy = 0
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]
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 \[x = \frac{\pi}{2}\]
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.
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The solution of the differential equation y1 y3 = y22 is
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 \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
`y=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 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 +(x-2 y)/ (2x- y)= 0`
The solution of `dy/dx + x^2/y^2 = 0` is ______
y2 dx + (xy + x2)dy = 0
Solve: `("d"y)/("d"x) + 2/xy` = x2
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
State whether the following statement is True or False:
The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
Solve the differential equation
`y (dy)/(dx) + x` = 0
Solve the differential equation
`x + y dy/dx` = x2 + y2
