Advertisements
Advertisements
प्रश्न
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
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 : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Solve the differential equation `dy/dx -y =e^x`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
x + y = tan–1y : y2 y′ + y2 + 1 = 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`
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 solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
Find the particular solution of the differential equation `(1+y^2)+(x-e^(tan-1 )y)dy/dx=` given that y = 0 when x = 1.
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
\[\frac{dy}{dx} = \left( x + y \right)^2\]
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
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:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]
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`
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.`
Solve: `2(y + 3) - xy "dy"/"dx"` = 0, given that y(1) = – 2.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The solution of differential equation coty dx = xdy is ______.
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
Which of the following differential equations has `y = x` as one of its particular solution?
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 ______.
