Advertisements
Advertisements
Question
Solve the differential equation:
`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1
Solution
`e^(x/y) (1- x/y) + (1 + e^(x/y)) dx/dy = 0`
`Put x = vy
`dx/dy = v + y "dv"/dy`
`:. e^v (1 - v) + (1 + e^v).(v + y "dv"/dy) = 0`
`v(1 + e^v) + y(1 + e^v). (dv)/dy = (v - 1)e^v`
`y(1 + e^v) (dv)/dy = e^v v - e^v - ve^v - v`
`y(1 + e^v) (dv)/(dy) = - (v + e^v)`
`(1 + e^v)/(-(v + e^v)) "dv"/dy = 1/y`
`-int (1 + e^v)/(v + e^v) dv = int 1/y dy`
`log c - log(v + e^v) = log y`
`c/(v + e^v) = y`
`y(v + e^v) = c`
`c = y(v + e^v)`
`c = y(x/y + e^(x"/"y))` ....(1)
when x = 0, y = 1
`c = 1(0 + e^o)`
c = 1
Put c = 1 in equation (1)
`1 = y(x/y + e^(x/y))`
APPEARS IN
RELATED QUESTIONS
The solution of the differential equation dy/dx = sec x – y tan x is:
(A) y sec x = tan x + c
(B) y sec x + tan x = c
(C) sec x = y tan x + c
(D) sec x + y tan x = c
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
The number of arbitrary constants in the particular solution of a differential equation of third order are ______.
Find the particular solution of the differential equation
`tan x * (dy)/(dx) = 2x tan x + x^2 - y`; `(tan x != 0)` given that y = 0 when `x = pi/2`
Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] is
The number of arbitrary constants in the particular solution of a differential equation of third order is
Which of the following differential equations has y = x as one of its particular solution?
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
`x cos x(dy)/(dx)+y(x sin x + cos x)=1`
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
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:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
