Advertisements
Advertisements
प्रश्न
General solution of `("d"y)/("d"x) + y` = sinx is ______.
Advertisements
उत्तर
General solution of `("d"y)/("d"x) + y` = sinx is y = `((sinx - cosx)/2) + "c"."e"^-x`.
Explanation:
The given differential equation is `("d"y)/("d"x) + y` = sinx
Since, it it a linear differential equation
∴ P = 1 and Q = sinx
Integrating factor I.F. = `"e"^(intPdx)`
= `"e"^(int1."d"x)`
= ex
∴ Solution is `y xx "i"."F". = int "Q" xx "I"."F". "D"x + "C"`
⇒ `y . "e"^x = int sin x . "e"6x "d"x + "c"` ....(1)
Let I = `int sin_"I"x . "e"_"II"^x "d"x`
I = `sin x . int "e"^x "d"x - int ("D"(sinx) . int"e"^x "d"x)"d"x`
I = `sinx . "e"^x - int cos_"I"x . "e"_"II"^x "d"x`
I = `sinx . "e"^x - [cosx . int "e"^x "d"x - int ("D"(cosx) int"e"^x "d"x)"d"x]`
I = `sin x . "e"6x - [cosx . "e"^x - int - sin x . "e"^x "d"x]`
I = `sin x . "e"^x - cos x . "e"^x - int sin x . "e"^x "d"x`
I = `sin x . "e"^x - cos x . "e"^x - "I"`
⇒ I + I = `"e"^x (sin x - cos x)`
⇒ 2I = `"e"^x (sinx - cosx)`
∴ I = `"e"^x/2 (sinx - cosx)`
From equation (1) we get
`y . "e"^x = "e"^x/2 (sinx - cosx) + "c"`
y = `((sinx - cosx)/2) + "c" . "e"^-x`
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`
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=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.
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
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 = cos x + C : y′ + sin x = 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 = 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.`
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
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 \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is
(x + y − 1) dy = (x + y) dx
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[\frac{dy}{dx} + 5y = \cos 4x\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
For the following differential equation, find a particular solution satisfying the given condition:
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]
For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
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 `"dy"/"dx" + y/x` = 1 is ______.
Find the general solution of `"dy"/"dx" + "a"y` = emx
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
Which of the following differential equations has `y = x` as one of its particular solution?
Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 0.
The curve passing through (0, 1) and satisfying `sin(dy/dx) = 1/2` is ______.
