Advertisements
Advertisements
प्रश्न
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
विकल्प
y = `"ce"^((-x^2)/2`
y = `"ce"^((x^2)/2`
y = `(x + "c")"e"^((x^2)/2`
y = `("c" - x)"e"^((x^2)/2`
Advertisements
उत्तर
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is y = `(x + "c")"e"^((x^2)/2`.
Explanation:
The given differential equation is `("d"y)/("d"x) = "e"^(x^2/2) + xy`
⇒ `("d"y)/("d"x) - xy = "e"^((x^2)/2`
Since it is linear differential equation
Where P = –x and Q = `"e"^((x^2)/2`
∴ Integrating factor I.F. = `"e"^(int Pdx)`
= `"e"^(int -x "d"x)`
= `"e"^(- x^2/2)`
So, the solution is `y xx "I"."F". = int "Q" xx "I"."F". "d"x + "c"`
⇒ `y xx "e"^( x^2/2) = int "e"^(x^2/2) "e"^(- x^2/2) "d"x + "c"`
⇒ `y xx "e"^(- x^2/2) = int "e"^0 "d"x + "c"`
⇒ `y xx "e"^(- x^2/2) = int 1 . "d"x + "c"`
⇒ `y xx "e"^(- x^2/2) = x + "c"`
∴ y = `(x + "c")"e"^(x^2/2)`.
APPEARS IN
संबंधित प्रश्न
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 differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 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.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x sin x : xy' = `y + x sqrt (x^2 - y^2)` (x ≠ 0 and x > y or x < -y)
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`
The number of arbitrary constants in the particular solution of a differential equation of third order are ______.
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
The number of arbitrary constants in the general solution of differential equation of fourth order is
x (e2y − 1) dy + (x2 − 1) ey dx = 0
(x + y − 1) dy = (x + y) dx
\[\frac{dy}{dx} - y \tan x = e^x\]
(x2 + 1) dy + (2y − 1) dx = 0
(x3 − 2y3) dx + 3x2 y dy = 0
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:- \[\frac{dy}{dx} + y = 1\]
For the following differential equation, find a particular solution satisfying the given condition:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
Find the equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`
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 differential equation of all non-horizontal lines in a plane.
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
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 solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
If the solution curve of the differential equation `(dy)/(dx) = (x + y - 2)/(x - y)` passes through the point (2, 1) and (k + 1, 2), k > 0, then ______.
Solve the differential equation:
`(xdy - ydx) ysin(y/x) = (ydx + xdy) xcos(y/x)`.
Find the particular solution satisfying the condition that y = π when x = 1.
