Advertisements
Advertisements
Question
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
Options
`("d"^2y)/("d"x^2) - alpha^2y` = 0
`("d"^2y)/("d"x^2) + alpha^2y` = 0
`("d"^2y)/("d"x^2) + alphay` = 0
`("d"^2y)/("d"x^2) - alphay` = 0
Advertisements
Solution
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is `("d"^2y)/("d"x^2) + alpha^2y` = 0.
Explanation:
Given equation is : y = A cos a x + B sin a x
Differentiating both sides w.r.t. x, we have
`("d"y)/("d"x) = -"A" sin alpha x * alpha + "B" cos alpha x * alpha`
= `- "A" alpha sin alphax + "B" alpha cos alpha x`
Again differentiating w.r.t. x, we get
`("d"^2y)/("d"x^2) = -"A"alpha^2 cos alpha x - "B" alpha^2 sin alpha x`
⇒ `("d"^2y)/("d"x^2) = -alpha^2 ("A" cos alphax + "B" sin alpha x)`
⇒ `("d"^2y)/("d"x^2) = - alpha^2y`
⇒ `("d"^2y)/("d"x^2) + alpha^2y` = 0
APPEARS IN
RELATED QUESTIONS
The differential equation of `y=c/x+c^2` is :
(a)`x^4(dy/dx)^2-xdy/dx=y`
(b)`(d^2y)/dx^2+xdy/dx+y=0`
(c)`x^3(dy/dx)^2+xdy/dx=y`
(d)`(d^2y)/dx^2+dy/dx-y=0`
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 general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y″ – y′ = 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:
xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`
Solve the differential equation `cos^2 x dy/dx` + y = tan x
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
\[\frac{dy}{dx} - y \tan x = e^x\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
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:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
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:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x2 + 1) dx, x ≠ 0.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
y = aemx+ be–mx satisfies which of the following differential equation?
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx is ______.
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
The member of arbitrary constants in the particulars solution of a differential equation of third order as
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
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 `x (dy)/(dx) = y(logy - logx + 1)`.
