Advertisements
Advertisements
प्रश्न
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
Advertisements
उत्तर
Given equation is `("d"y)/("d"x) -3y = sin2x`
Here, P = –3 and Q = sin2x
∴ Integrating factor I.F. = `"e"^(int Pdx)`
= `"e"^(int-3dx)`
= `"e"^(-3x)`
∴ Solution is `y xx "I"."F". = int "Q" . "I"."F". "d"x + "c"`
⇒ `y . "e"^(-3x) = int sin2x . "e"^(-3x) "d"x + "c"`
Let I = `int sin_"I" 2x . "e"_"II"^(-3x) "d"x`
⇒ I = `sin 2x . int "e"^(-3x)"d"x - int("D"(sin 2x) . int"e"^(-3x) "d"x)"d"x`
⇒ I = `sin 2x . "e"^(-3x)/(-3) - int 2 cos2x . "e"^(-3x)/(-3) "d"x`
⇒ I = `"e"^(-3x)/(-3) sin2x + 2/3 int cos_"I" 2x . "e"_"II"^(-3x) "d"x`
⇒ I = `"e"^(-3x)/(-3) sin 2x + 2/3 [cos 2x . int "e"^(-3x) "d"x - int["D" cos2x . int "e"^(-3x) "d"x]"d"x]`
⇒ I = `"e"^(-3x)/(-3) sin 2x + 2/3 [cos 2x . "e"^(-3x)/(-3) - 2sin 2x . "e"^(-3x)/(-3)]"d"x`
⇒ I = `"e"^(-3x)/(-3) sin 2x - 2/9 cos2x . "e"^(-3x) - 4/9 int sin 2x. "e"^(-3x) "d"x`
⇒ `"e"^(-3x)/(-3) sin2x - 2/9 "e"^(-3x) cos 2x - 4/9 "I"`
⇒ `"I" + 4/9 "I" = "e"^(-3x)/(-3) sin 2x - 2/9 "e"^(-3x) cos 2x`
⇒ `13/9 "I" = - 1/9 [3"e"^(-3x) sin2x + 2"e"^(-3x) cos2x]`
⇒ I = `- 1/13 "e"^(-3x) [3 sin 2x + 2 cos2x]`
∴ The equation becomes `y . "e"^(-3x) = - 1/13 "e"^(-3x) [3 sin 2x + 2 cos 2x] + "c"`
∴ y = `- 1/13 [3 sin 2x + 2 cos 2x] + "c" . "e"^(3x)`
Hence, the required solution is y = `-[(3sin2x + 2cos2x)/13] + "c" . "e"^(3x)`
APPEARS IN
संबंधित प्रश्न
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
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 – cos y = x : (y sin y + cos y + x) y′ = y
Solve the differential equation `cos^2 x dy/dx` + y = tan x
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
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`
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
The general solution of the differential equation \[\frac{dy}{dx} + y\] g' (x) = g (x) g' (x), where g (x) is a given function of x, is
Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] is
If m and n are the order and degree of the differential equation \[\left( y_2 \right)^5 + \frac{4 \left( y_2 \right)^3}{y_3} + y_3 = x^2 - 1\], then
The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
(x + y − 1) dy = (x + y) dx
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sin^{- 1} x\]
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:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` 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 solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The member of arbitrary constants in the particulars solution of a differential equation of third order as
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
