Advertisements
Advertisements
Question
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]
Advertisements
Solution
We have,
\[\frac{dy}{dx} = \sqrt{4 - y^2}\]
\[ \Rightarrow \frac{1}{\sqrt{4 - y^2}}dy = dx\]
Integrating both sides, we get
\[\int\frac{1}{\sqrt{4 - y^2}}dy = \int dx\]
\[ \Rightarrow \sin^{- 1} \frac{y}{2} = x + C\]
\[ \Rightarrow \frac{y}{2} = \sin \left( x + C \right)\]
\[ \Rightarrow y = 2\sin \left( x + C \right)\]
APPEARS IN
RELATED QUESTIONS
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=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
The number of arbitrary constants in the particular solution of a differential equation of third order are ______.
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
Solve the differential equation `cos^2 x dy/dx` + y = tan x
Solve the differential equation:
`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1
The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
Which of the following differential equations has y = x as one of its particular solution?
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sin^{- 1} x\]
Solve the following differential equation:- \[x \cos\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 2y = \sin x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Find the differential equation of all non-horizontal lines in a plane.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
Find the general solution of the differential equation:
`log((dy)/(dx)) = ax + by`.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
