Advertisements
Advertisements
प्रश्न
Find the general solution of `"dy"/"dx" + "a"y` = emx
Advertisements
उत्तर
Given equation is `"dy"/"dx" + "a"y` = emx
Here, P = a and Q = emx
∴ I.F. = `"e"^(int Pdx)`
= `"e"^(int a .dx)`
= eax.
Solution of equation is `y xx "I"."F" = int "Q" "I"."F" "d"x + "c"`
⇒ `y."e"^("a"x) = int "e"^"mx" . "e"^("a"x) "d"x + "c"`
⇒ `y . "e"^("a"x) = int "e"^(("m" + "a")x) "d"x + "c"`
⇒ `y . "e"^("a"x) = "e"^(("m" + "a")x)/(("m" + "a")) + "c"`
⇒ y = `"e"^(("m" + "a")x)/(("m" + "a")) . "e"^(-"a"x) + "c"."e"^(-"a"x)`
∴ y = `"e"^("m"x)/(("m" + "a")) + "c" . "e"^(-"a"x)`
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
Find the differential equation representing the curve y = cx + c2.
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
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
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 dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
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 = Ax : xy′ = y (x ≠ 0)
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)`
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.
Write the order of the differential equation associated with the primitive y = C1 + C2 ex + C3 e−2x + C4, where C1, C2, C3, C4 are arbitrary constants.
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]
\[\frac{dy}{dx} + 2y = \sin 3x\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
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`
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 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.
If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 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 ______.
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` 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:
`log((dy)/(dx)) = ax + by`.
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
