Advertisements
Advertisements
प्रश्न
उत्तर
We have,
\[x\frac{dy}{dx} + \cot y = 0\]
\[ \Rightarrow x\frac{dy}{dx} = - \cot y\]
\[ \Rightarrow \frac{1}{x}dx = - \frac{1}{\cot y}dy\]
\[ \Rightarrow \frac{1}{x}dx = - \tan y dy\]
Integrating both sides, we get
\[\int\frac{1}{x}dx = - \int\tan y dy\]
\[ \Rightarrow \ln \left| x \right| = - \ln\left| \sec y \right| + \ln C\]
\[ \Rightarrow \ln \left| x \right| = \ln \left| \cos y \right| + \ln C\]
\[ \Rightarrow x = C \cos y \]
\[\text{ Hence, }x = C \cos y\text{ is the required solution .} \]
APPEARS IN
संबंधित प्रश्न
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
(sin x + cos x) dy + (cos x − sin x) dx = 0
(y2 − 2xy) dx = (x2 − 2xy) dy
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
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 which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]
The x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| xy = log y + k | y' (1 - xy) = y2 |
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
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
State whether the following is True or False:
The integrating factor of the differential equation `dy/dx - y = x` is e-x
Select and write the correct alternative from the given option for the question
Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in
For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve: ydx – xdy = x2ydx.
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
Solve the differential equation
`y (dy)/(dx) + x` = 0
