Advertisements
Advertisements
Question
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
Options
y =`"e"^(x - y) = x^2 "e"^-y + "c"`
`"e"^y - "e"^x = x^3/3 + "c"`
`"e"^x + "e"^y = x^3/3 + "c"`
`"e"^x - "e"^y = x^3/3 + "c"`
Advertisements
Solution
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is `"e"^y - "e"^x = x^3/3 + "c"`.
Explanation:
The given differential equation is `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y`
⇒ `("d"y)/("d"x) = "e"^x . "e"^-y + x^2 . "e"^-y`
⇒ `("d"y)/("d"x) = "e"^-y ("e"^x + x^2)`
⇒ `("d"y)/"e"^-y = ("e"^x + x^2)"d"x`
⇒ `"e"^y . "d"y = ("e"^x + x^2)"d"x`
Integrating both sides, we have
`int "e"^x "d"y = int ("e"^x + x^2) "d"x`
⇒ `"e"^y = "e"^x + x^3/3 + "c"`
⇒ `"e"^y - "e"^x = x^3/3 + "c"`
APPEARS IN
RELATED QUESTIONS
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 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 = ex + 1 : y″ – y′ = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`
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 = 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:
`y = sqrt(a^2 - x^2 ) x in (-a,a) : x + y dy/dx = 0(y != 0)`
Solve the differential equation `cos^2 x dy/dx` + y = tan x
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
(1 + y + x2 y) dx + (x + x3) dy = 0
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
For the following differential equation, find a particular solution satisfying the given condition:
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
Solution of differential equation xdy – ydx = 0 represents : ______.
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The member of arbitrary constants in the particulars solution of a differential equation of third order as
Which of the following differential equations has `y = x` as one of its particular solution?
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
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.
