Advertisements
Advertisements
प्रश्न
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
विकल्प
xy = – ex
xy = – e-x
xy = – 1
y = 2ex – 1
उत्तर
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by y = 2ex – 1.
Explanation:
The given differential equation is `("d"y)/("d"x) - y` = 1
Here, P = –1, Q = 1
∴ Integrating factor, I.F. = `"e"^(intPdx)`
= `"e"^(int -1"d"x)`
= `"e"^-x`
So, the solution is `y xx "I"."F". = int "Q" ."I"."F". "d"x + "c"`
⇒ `y xx "e"^-x = int 1."e"^-x "d"x + "c"`
⇒ `y * "e"^-x = -"e"^-x + "c"`
Put x = 0, y = 1
⇒ `1. "e"^0 = - "e"^0 + "c"`
⇒ 1 = `-1 + "c"`
∴ c = 2
So the equation is `y * "e"^-x = -"e"^-x + 2`
⇒ y = `-1 + 2"e"^x`
= `2"e"^x - 1`.
APPEARS IN
संबंधित प्रश्न
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
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`
Find the particular solution of the differential equation `(1+x^2)dy/dx=(e^(mtan^-1 x)-y)` , give that y=1 when x=0.
Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.
Solve the differential equation `dy/dx -y =e^x`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 0
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
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`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
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`
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 general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
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.
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
(1 + y + x2 y) dx + (x + x3) dy = 0
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\frac{dy}{dx} + 5y = \cos 4x\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sin^{- 1} x\]
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
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:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Find the general solution of `"dy"/"dx" + "a"y` = emx
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
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 ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 0.
