Advertisements
Advertisements
Question
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
Options
`"e"^(x^2 - y)` = c
`"e"^-y + "e"^(x^2)` = c
`"e"^-y = "e"^(x^2)` + c
`"e"^(x^2 + y)` = c
Advertisements
Solution
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is `"e"^-y = "e"^(x^2)` + c.
Explanation:
The given differential equation is `("d"y)/("d"x) = 2x"e"^(x^2 - y)`
⇒ `("d"y)/("d"x) = 2x . "e"^(x^2) . "e"^-y`
⇒ `("d"y)/("e"^-y) = 2x . "e"^(x^2) "d"x`
Integrating both sides, we have
`int ("d"y)/("e"^-y) = int 2x . "e"^(x^2) "d"x`
⇒ `int "e"^y "d"y = int 2x . "e"^(x^2) "d"x`
Pit in R.H.S. x2 = t
∴ 2x dx = dt
∴ `int "e"^y "d"y = int "e"^"t" "dt"`
⇒ ey = et + c
⇒ ey = `"e"^(y^2) + "c"`
APPEARS IN
RELATED QUESTIONS
The differential equation of `y=c/x+c^2` is :
(a)`x^4(dy/dx)^2-xdy/dx=y`
(b)`(d^2y)/dx^2+xdy/dx+y=0`
(c)`x^3(dy/dx)^2+xdy/dx=y`
(d)`(d^2y)/dx^2+dy/dx-y=0`
If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`
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 the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
Find the particular solution of the differential equation x (1 + y2) dx – y (1 + x2) dy = 0, given that y = 1 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 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)
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
If y = etan x+ (log x)tan x then find dy/dx
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`
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The number of arbitrary constants in the general solution of differential equation of fourth order is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
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:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x2 + 1) dx, x ≠ 0.
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.`
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
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.
