Advertisements
Advertisements
प्रश्न
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
विकल्प
log y = kx
y = kx
xy = k
y = k log x
Advertisements
उत्तर
y = kx
We have,
\[\frac{dy}{dx} = \frac{y}{x}\]
\[ \Rightarrow \frac{1}{y}dy = \frac{1}{x}dx\]
Integrating both sides, we get
\[\int\frac{1}{y}dy = \int\frac{1}{x}dx\]
\[ \Rightarrow \log y = \log x + \log k\]
\[ \Rightarrow \log y - \log x = \log k\]
\[ \Rightarrow \log\left( \frac{y}{x} \right) = \log k\]
\[ \Rightarrow \frac{y}{x} = k\]
\[ \Rightarrow y = kx\]
APPEARS IN
संबंधित प्रश्न
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Find the differential equation representing the curve y = cx + c2.
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
If y = etan x+ (log x)tan x then find dy/dx
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
The number of arbitrary constants in the general solution of differential equation of fourth order is
The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is
\[\frac{dy}{dx} + 1 = e^{x + y}\]
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
\[\frac{dy}{dx} - y \tan x = e^x\]
(x2 + 1) dy + (2y − 1) dx = 0
x2 dy + (x2 − xy + y2) dx = 0
\[\frac{dy}{dx} + y = 4x\]
\[\frac{dy}{dx} + 5y = \cos 4x\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 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(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.
The general solution of ex cosy dx – ex siny dy = 0 is ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
The value of c in the particular solution given that y(0) = 0 and k = 0.049 is ______.
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
