Advertisements
Advertisements
Question
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
Options
\[\phi\left( \frac{y}{x} \right) = kx\]
\[x\phi\left( \frac{y}{x} \right) = k\]
\[\phi\left( \frac{y}{x} \right) = ky\]
\[y\phi\left( \frac{y}{x} \right) = k\]
Advertisements
Solution
\[\phi\left( \frac{y}{x} \right) = kx\]
We have,
\[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\]
Let y = vx
\[ \Rightarrow \frac{dy}{dx} = v + x\frac{dv}{dx}\]
\[ \therefore v + x\frac{dv}{dx} = v + \frac{\phi\left( v \right)}{\phi'\left( v \right)}\]
\[ \Rightarrow x\frac{dv}{dx} = \frac{\phi\left( v \right)}{\phi'\left( v \right)}\]
\[ \Rightarrow \frac{\phi\left( v \right)}{\phi'\left( v \right)}dv = \frac{1}{x}dx\]
Integrating both sides, we get
\[\int\frac{\phi'\left( v \right)}{\phi\left( v \right)}dv = \int\frac{1}{x}dx\]
\[ \Rightarrow \log \left| \phi\left( v \right) \right| = \log \left| x \right| + \log k\]
\[ \Rightarrow \log \left| \phi\left( \frac{y}{x} \right) \right| - \log \left| x \right| = \log k\]
\[ \Rightarrow \log\left| \frac{\phi\left( \frac{y}{x} \right)}{x} \right| = \log k\]
\[ \Rightarrow \frac{\phi\left( \frac{y}{x} \right)}{x} = k\]
\[ \Rightarrow \phi\left( \frac{y}{x} \right) = kx\]
APPEARS IN
RELATED QUESTIONS
The differential equation of the family of curves y=c1ex+c2e-x is......
(a)`(d^2y)/dx^2+y=0`
(b)`(d^2y)/dx^2-y=0`
(c)`(d^2y)/dx^2+1=0`
(d)`(d^2y)/dx^2-1=0`
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (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)`
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
Solve the differential equation `cos^2 x dy/dx` + y = tan x
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Find the particular solution of the differential equation `(1+y^2)+(x-e^(tan-1 )y)dy/dx=` given that y = 0 when x = 1.
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
(x2 + 1) dy + (2y − 1) dx = 0
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\frac{dy}{dx} + y = 4x\]
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:- \[\frac{dy}{dx} + y = 1\]
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
Solution of differential equation xdy – ydx = 0 represents : ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The solution of differential equation coty dx = xdy is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
Find the general solution of the differential equation:
`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
