Advertisements
Advertisements
Question
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
Options
log y = kx
y = kx
xy = k
y = k log x
Advertisements
Solution
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
RELATED QUESTIONS
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 = ex + 1 : y″ – y′ = 0
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
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The general solution of the differential equation \[\frac{dy}{dx} + y\] g' (x) = g (x) g' (x), where g (x) is a given function of x, is
Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
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
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
cos (x + y) dy = dx
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
(x2 + 1) dy + (2y − 1) dx = 0
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
Solve the following differential equation:-
\[\frac{dy}{dx} - y = \cos x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x2 + 1) dx, x ≠ 0.
Solve the differential equation: `(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = "x"^2 + 2`
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.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The differential equation for which y = acosx + bsinx is a solution, 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.
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
