Advertisements
Advertisements
Question
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
Advertisements
Solution
We have,
\[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2 \sin^2 \frac{x}{2}}{2 \cos^2 \frac{x}{2}}\]
\[ \Rightarrow \frac{dy}{dx} = \tan^2 \frac{x}{2}\]
\[ \Rightarrow dy = \left( \tan^2 \frac{x}{2} \right)dx\]
Integrating both sides, we get
\[\int dy = \int\left( \tan^2 \frac{x}{2} \right)dx\]
\[ \Rightarrow \int dy = \int\left( \sec^2 \frac{x}{2} - 1 \right)dx\]
\[ \Rightarrow y = 2 \tan \frac{x}{2} - x + C\]
APPEARS IN
RELATED QUESTIONS
The differential equation of `y=c/x+c^2` is :
(a)`x^4(dy/dx)^2-xdy/dx=y`
(b)`(d^2y)/dx^2+xdy/dx+y=0`
(c)`x^3(dy/dx)^2+xdy/dx=y`
(d)`(d^2y)/dx^2+dy/dx-y=0`
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = cos x + C : y′ + sin x = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y = sqrt(a^2 - x^2 ) x in (-a,a) : x + y dy/dx = 0(y != 0)`
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
If y = etan x+ (log x)tan x then find dy/dx
Write the order of the differential equation associated with the primitive y = C1 + C2 ex + C3 e−2x + C4, where C1, C2, C3, C4 are arbitrary constants.
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is
The number of arbitrary constants in the particular solution of a differential equation of third order is
\[\frac{dy}{dx} = \left( x + y \right)^2\]
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sin^{- 1} x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
Solve the following differential equation:-
y dx + (x − y2) dy = 0
Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.
If y(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
The solution of differential equation coty dx = xdy is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
