Advertisements
Advertisements
Question
Advertisements
Solution
We have,
\[\frac{dy}{dx} = \sin^2 y\]
\[ \Rightarrow \frac{dx}{dy} = \frac{1}{\sin^2 y}\]
\[ \Rightarrow dx = {cosec}^2 y dy\]
Integrating both sides, we get
\[\int dx = \int {cosec}^2 y dy\]
\[ \Rightarrow x = - \cot y + C\]
\[ \Rightarrow x + \cot y = C\]
\[\text{ Hence, }x + \cot y = \text{ C is the required solution }.\]
APPEARS IN
RELATED QUESTIONS
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
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.
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 |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = ex
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
xy (y + 1) dy = (x2 + 1) dx
y (1 + ex) dy = (y + 1) ex dx
(y + xy) dx + (x − xy2) dy = 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 = \[\frac{\pi}{2}\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.
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 \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
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.
Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
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
Solve the differential equation xdx + 2ydy = 0
Solve the following differential equation y2dx + (xy + x2) dy = 0
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve the differential equation `"dy"/"dx" + 2xy` = y
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
