Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
We have,
\[\frac{dy}{dx} + \frac{\cos x \sin y}{\cos y} = 0\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{\cos x \sin y}{\cos y}\]
\[ \Rightarrow \frac{\cos y}{\sin y}dy = - \cos x dx\]
\[ \Rightarrow \cot y dy = - \cos x dx\]
Integrating both sides, we get
\[\int\cot y dy = - \int\cos x dx\]
\[ \Rightarrow \log \left| \sin y \right| = - \sin x + C\]
\[\text{ Hence,} \log \left| \sin y \right| = - \sin x +\text{ C is the required solution .} \]
APPEARS IN
संबंधित प्रश्न
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
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 |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
(y + xy) dx + (x − xy2) dy = 0
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
3x2 dy = (3xy + y2) dx
Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
The slope of the tangent at each point of a curve is equal to the sum of the coordinates of the point. Find the curve that passes through the origin.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).
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 |
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
Solve the following differential equation.
`xy dy/dx = x^2 + 2y^2`
Solve
`dy/dx + 2/ x y = x^2`
y dx – x dy + log x dx = 0
Solve the differential equation `("d"y)/("d"x) + y` = e−x
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 function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
