Advertisements
Advertisements
Question
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
Advertisements
Solution
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
RELATED QUESTIONS
Find the differential equation representing the curve y = cx + c2.
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 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 = x sin x : xy' = `y + x sqrt (x^2 - y^2)` (x ≠ 0 and x > y or x < -y)
If y = etan x+ (log x)tan x then find dy/dx
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
The solution of the differential equation (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is
(x3 − 2y3) dx + 3x2 y dy = 0
\[\frac{dy}{dx} + 5y = \cos 4x\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]
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:- \[x \cos\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Solve the following differential equation:-
\[\left( x + y \right)\frac{dy}{dx} = 1\]
Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
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.
Find the general solution of the differential equation:
`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`
