Advertisements
Advertisements
Question
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
Options
sin x
sec x
tan x
cos x
Solution
sec x
We have,
\[\cos x\frac{dy}{dx} + y \sin x = 1\]
Dividing both sides by cos x, we get
\[\frac{dy}{dx} + \frac{\sin x}{\cos x}y = \frac{1}{\cos x}\]
\[ \Rightarrow \frac{dy}{dx} + \left( \tan x \right)y = \frac{1}{\cos x}\]
\[\text{Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = \tan x\]
\[Q = \frac{1}{\cos x}\]
Now,
\[ I . F . = e^{\int\tan xdx} = e^{log\left( \sec x \right)} \]
\[ = \sec x\]
APPEARS IN
RELATED QUESTIONS
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.
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
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 = AeBx is a solution of the differential equation
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 |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
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 \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
Differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex
(1 + x2) dy = xy dx
x cos y dy = (xex log x + ex) dx
y (1 + ex) dy = (y + 1) ex dx
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
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.
If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).
(x + 2y) dx − (2x − y) dy = 0
A population grows at the rate of 5% per year. How long does it take for the population to double?
In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?
Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.
A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
The differential equation satisfied by ax2 + by2 = 1 is
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
Solve the following differential equation.
`xy dy/dx = x^2 + 2y^2`
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
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.
`xy dy/dx = x^2 + 2y^2`
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
