Advertisements
Advertisements
प्रश्न
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
Advertisements
उत्तर
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by `x"e"^(intPdx) = int "Q"_1"e"^(int P_1"d"y) "d"y + "C"`.
Explanation:
We have `("d"x)/("d"x) + "P"_1x = "Q"_1`
For solving such equation we multiply both sides by
Integrating factor = I.F. = `"e"^(int Pdx)`
So we get `"e"^(intPdx) (("d"x)/("d"y) + "P"_1x) = "Q"_1"e"^(intPdx)`
⇒ `("d"x)/("d"y) "e"^(intPdx) + "P"_1"e"^(intPdy) = "Q"_1"e"^(intP_1dy)`
⇒ `"d"/("d"y)(x"e"^(intP_1dy)) = "Q"_1"e"^(intP_1dy)`
⇒ `int "d"/("d"y) (x"e"^(intP_1dy))"d"y = int "Q"_1"e"^(intP_1dy) "d"y`
⇒ `x"e"^(intP_1"d"y) = int"Q"_1"e"^(intP_1dy) "d"y + "C"`
This is the required solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
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 log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y – cos y = x : (y sin y + cos y + x) y′ = y
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
If y = etan x+ (log x)tan x then find dy/dx
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Solve the differential equation (x2 − yx2) dy + (y2 + x2y2) dx = 0, given that y = 1, when x = 1.
x (e2y − 1) dy + (x2 − 1) ey dx = 0
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[\frac{dy}{dx} + y = 4x\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[\frac{dy}{dx} - y = \cos x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
Find the differential equation of all non-horizontal lines in a plane.
The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.
Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.
Solution of differential equation xdy – ydx = 0 represents : ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
