Advertisements
Advertisements
प्रश्न
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
पर्याय
circles
straight lines
ellipses
parabolas
Advertisements
उत्तर
parabolas
We have,
\[2x\frac{dy}{dx} - y = 3\]
\[ \Rightarrow 2x\frac{dy}{dx} = 3 + y\]
\[ \Rightarrow \frac{1}{3 + y}dy = \frac{1}{2x}dx\]
Integrating both sides, we get
\[\int\frac{1}{3 + y}dy = \frac{1}{2}\int\frac{1}{x}dx\]
\[ \Rightarrow \log \left| 3 + y \right| = \frac{1}{2}\log \left| x \right| + \log C\]
\[ \Rightarrow \log \left| 3 + y \right| - \log \left| x^\frac{1}{2} \right| = \log C\]
\[ \Rightarrow \log \left| \frac{3 + y}{\sqrt{x}} \right| = \log C\]
\[ \Rightarrow \frac{3 + y}{\sqrt{x}} = C\]
\[ \Rightarrow 3 + y = C\sqrt{x}\]
Squaring both sides, we get
\[ \left( 3 + y \right)^2 = Cx . . . . . \left( 1 \right)\]
\[\text{ Thus, }\left( 1 \right)\text{ represents the equation of parabolas .}\]
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
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`
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y – cos y = x : (y sin y + cos y + x) y′ = y
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
The number of arbitrary constants in the particular solution of a differential equation of third order is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
(x3 − 2y3) dx + 3x2 y dy = 0
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
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:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
Solve the following differential equation:-
y dx + (x − y2) dy = 0
Find the equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`
Find the differential equation of all non-horizontal lines in a plane.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
