Advertisements
Advertisements
Question
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
Options
ex
log x
log (log x)
x
Advertisements
Solution
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is log x.
Explanation:
Given equation can be written as `"dy"/"dx" + y/(xlogx) = 2/x`.
Therefore, I.F. = `"e"^(int 1/(xlogx) "d"x)`
= `"e"^(log (logx))`
= log x.
APPEARS IN
RELATED QUESTIONS
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x + y\frac{dy}{dx} = 0\]
|
\[y = \pm \sqrt{a^2 - x^2}\]
|
xy dy = (y − 1) (x + 1) dx
(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:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
(x2 − y2) dx − 2xy dy = 0
3x2 dy = (3xy + y2) dx
Solve the following initial value problem:-
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]
The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
Find the equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\] at any point (x, y) on it.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
The differential equation `y dy/dx + x = 0` represents family of ______.
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Solve the following differential equation.
`dy/dx + 2xy = x`
Solve the following differential equation.
`(x + a) dy/dx = – y + a`
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` 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.
y2 dx + (xy + x2)dy = 0
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
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
Solve the following differential equation
sec2 x tan y dx + sec2 y tan x dy = 0
Solution: sec2 x tan y dx + sec2 y tan x dy = 0
∴ `(sec^2x)/tanx "d"x + square` = 0
Integrating, we get
`square + int (sec^2y)/tany "d"y` = log c
Each of these integral is of the type
`int ("f'"(x))/("f"(x)) "d"x` = log |f(x)| + log c
∴ the general solution is
`square + log |tan y|` = log c
∴ log |tan x . tan y| = log c
`square`
This is the general solution.
Solve the differential equation
`x + y dy/dx` = x2 + y2
