Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
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
संबंधित प्रश्न
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Show that y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
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
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
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 equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\] at any point (x, y) on it.
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
The differential equation satisfied by ax2 + by2 = 1 is
y2 dx + (x2 − xy + y2) dy = 0
Solve the differential equation:
`"x"("dy")/("dx")+"y"=3"x"^2-2`
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
The solution of `dy/ dx` = 1 is ______.
Solve:
(x + y) dy = a2 dx
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
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 = ex is solution ______ of differential equation
Solve: ydx – xdy = x2ydx.
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha - y cos alpha` = 0, then the value of `a^2 + b^2` is
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
