Advertisements
Advertisements
प्रश्न
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
पर्याय
x
ex
log x
log (log x)
Advertisements
उत्तर
log x
We have,
\[\left( x \log x \right)\frac{dy}{dx} + y = 2 \log x\]
Dividing both sides by (x log x) we get,
\[\frac{dy}{dx} + \frac{y}{x \log x} = 2\frac{\log x}{x \log x}\]
\[ \Rightarrow \frac{dy}{dx} + \frac{y}{x \log x} = \frac{2}{x}\]
\[ \Rightarrow \frac{dy}{dx} + \left( \frac{1}{x \log x} \right)y = \frac{2}{x}\]
\[\text{ Comparing with }\frac{dy}{dx} + Py = Q \text{ we get, }\]
\[P = \frac{1}{x \log x} \text{ and }Q = \frac{2}{x}\]
\[\text{ Now, }I . F = e^{\int P\ dx} = e^{\int\frac{1}{x\log x}dx} \]
\[ = e^{log\left( \log x \right)} \]
\[ = \log x\]
APPEARS IN
संबंधित प्रश्न
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Show that y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
Verify that y = cx + 2c2 is a solution of the differential equation
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 \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 3\] Function y = ex + e2x
xy dy = (y − 1) (x + 1) dx
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Solve the following initial value problem:-
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]
A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.
The slope of the tangent at a point P (x, y) on a curve is \[\frac{- x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.
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.
Define a differential equation.
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Show that y = ae2x + be−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
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` |
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
Form the differential equation from the relation x2 + 4y2 = 4b2
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.
(x2 − y2 ) dx + 2xy dy = 0
Solve the following differential equation.
`dy/dx + y` = 3
The solution of `dy/ dx` = 1 is ______.
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Solve the differential equation `"dy"/"dx" + 2xy` = y
