Advertisements
Advertisements
Question
Solution
We have,
\[\cos x \cos y \frac{dy}{dx} = - \sin x \sin y \]
\[ \Rightarrow \frac{\cos y}{\sin y}dy = \frac{- \sin x}{\cos x}dx\]
\[ \Rightarrow \cot y\ dy = - \tan x\ dx\]
Integrating both sides, we get
\[\int \cot y\ dy = - \int\tan x\ dx\]
\[ \Rightarrow \log \left| \sin y \right| = - \log \left| \sec x \right| + \log C\]
\[ \Rightarrow \log \left| \sin y \right| = \log \left| \cos x \right| + \log C\]
\[ \Rightarrow \sin y = C \cos x\]
\[\text{ Hence, }\sin y = C \cos x\text{ is the required solution . }\]
APPEARS IN
RELATED QUESTIONS
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
(1 − x2) dy + xy dx = xy2 dx
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
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.
Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
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 |
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Solve the following differential equation.
x2y dx − (x3 + y3 ) dy = 0
Solve the following differential equation.
`dy/dx + 2xy = x`
Solve the following differential equation.
dr + (2r)dθ= 8dθ
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.
Solve the differential equation:
`e^(dy/dx) = 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
Solve the differential equation xdx + 2ydy = 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.
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
Solve: ydx – xdy = x2ydx.
