Advertisements
Advertisements
Question
For the given below, verify that the given function (implicit or explicit) is a solution to the corresponding differential equation.
`y = e^x (acos x + b sin x) : (d^2y)/(dx^2) - 2 dy/dx + 2y = 0`
Advertisements
Solution
Given function y = ex (a cos x + b sin x)
On differentiating
`dy/dx = e^x (a sin x + b cos x) + e^x (a cos x + b sin x)`
`= e^x [(a + b) cos x + (b - a) sin x]`
On differentiating again,
`(d^2y)/dx^2 = e^x [- (a + b) sin x + (b - a) cos x] + e^x [(a + b) cos x + (b - a) sin x]`
`(d^2y)/(dx^2)= e^x [2b cos x - 2a sin x]`
L.H.S. ⇒ `(d^2y)/(dx^2) - 2 dy/dx + 2y`
`= e^x [2b cos x - 2a sin x] - 2e^x [(a + b) cos x + (b - a) sin x] + 2e^x [a cos x + b sin x] = 0`
`= e^x [(2b - 2a - 2b + 2a)] cos x + (- 2a - 2b + 2a + 2b) sin x`
`= "e"^x xx 0 + 0 xx sin x = 0` R.H.S.
Hence the given function is a solution to the differential equation.
APPEARS IN
RELATED QUESTIONS
Order and degree of the differential equation `[1+(dy/dx)^3]^(7/3)=7(d^2y)/(dx^2)` are respectively
(A) 2, 3
(B) 3, 2
(C) 7, 2
(D) 3, 7
Determine the order and degree (if defined) of the differential equation:
`(d^2y)/(dx^2)^2 + cos(dy/dx) = 0`
The degree of the differential equation `((d^2y)/(dx^2))^3 + ((dy)/(dx))^2 + sin ((dy)/(dx)) + 1 = 0` is ______.
Write the degree of the differential equation \[x^3 \left( \frac{d^2 y}{d x^2} \right)^2 + x \left( \frac{dy}{dx} \right)^4 = 0\]
Write the degree of the differential equation \[\left( 1 + \frac{dy}{dx} \right)^3 = \left( \frac{d^2 y}{d x^2} \right)^2\]
The degree of the differential equation \[\left( \frac{d^2 y}{d x^2} \right)^2 - \left( \frac{dy}{dx} \right) = y^3\], is
The order of the differential equation \[2 x^2 \frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + y = 0\], is
Determine the order and degree of the following differential equation:
`root(3)(1 +("dy"/"dx")^2) = ("d"^2"y")/"dx"^2`
Determine the order and degree of the following differential equation:
`(dy)/(dx) = (2sin x + 3)/(dy/dx)`
Determine the order and degree of the following differential equation:
(y''')2 + 3y'' + 3xy' + 5y = 0
Determine the order and degree of the following differential equation:
`"dy"/"dx" = 3"y" + root(4)(1 + 5 ("dy"/"dx")^2)`
Determine the order and degree of the following differential equations.
`(d^4y)/dx^4 + [1+(dy/dx)^2]^3 = 0`
State whether the following statement is true or false:
Order and degree of a differential equation are always positive integers.
State the degree of differential equation `e^((dy)/(dx)) + (dy)/(dx)` = x
Order and degree of differential equation`(("d"^3y)/("d"x^3))^(1/6)`= 9 is ______
The degree of the differential equation `("d"^4"y")/"dx"^4 + sqrt(1 + ("dy"/"dx")^4)` = 0 is
The order of the differential equation whose general solution is given by `y=C_(1)e^(2x+C_2)+C_3e^x+C_4sin(x+C_5)` is ______.
The order of the differential equation of all circles of radius r, having centre on X-axis and passing through the origin is ______.
Order of the differential equation representing the family of parabolas y2 = 4ax is ______.
The degree of the differential equation `("dy"/"dx")^2 + (("d"^2y)/("d"x^2))^2` = 0 is ______.
The degree of the differential equation `[1 + (("d"y)/("d"x))^2]^(3/2) = ("d"^2y)/("d"x^2)` is ______.
The degree of the differential equation `("d"^2"y")/("dx"^2) + 3("dy"/"dx")^2 = "x"^2 (("d"^2"y")/("dx"^2))^2` is:
Write the sum of the order and the degree of the following differential equation:
`d/(dx) (dy/dx)` = 5
y2 = (x + c)3 is the general solution of the differential equation ______.
The order of the differential equation of all parabolas, whose latus rectum is 4a and axis parallel to the x-axis, is ______.
The order and degree of the differential eqµation whose general solution is given by `(d^2y)/(dx^2) + (dy/dx)^50` = In `((d^2y)/dx^2)` respectively, are ______.
The sum of the order and the degree of the differential equation `d/dx[(dy/dx)^3]` is ______.
The degree of the differential equation `((d^3y)/(dx^2))^4 + ((d^2y)/(dx^2))^5 + (dy)/(dx) + y = 0` is ______.
What is the order of the differential equation \[\frac{dy}{dx} - \cos x = 0\]?
What is the degree of the differential equation \[\frac{dy}{dx} - \cos x = 0\]?
Based on the differential equation illustrated with \[\left(\frac{d^{3}y}{dx^{3}}\right)^{2} + 5\frac{dy}{dx} = \sin(x)\], what is the order?
Based on the differential equation \[\left(\frac{d^{3}y}{dx^{3}}\right)^{2} + 5\frac{dy}{dx} = \sin(x)\], what is the degree?
Which of the following statements is true regarding order and degree of differential equations?
Why is the degree of the differential equation \[y'' + \sin(y') = 0\] not defined?
What is the key difference between order and degree of a differential equation?
