Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Write the degree of the differential equation `x^3((d^2y)/(dx^2))^2+x(dy/dx)^4=0`
Determine the order and degree (if defined) of the differential equation:
y′ + y = ex
For the given below, verify that the given function (implicit or explicit) is a solution to the corresponding differential equation.
`y = xsin 3x : (d^2y)/(dx^2) + 9y - 6 cos 3x = 0`
Write the order and degree of the differential equation
\[y = x\frac{dy}{dx} + a\sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]
Write the degree of the differential equation
\[\frac{d^2 y}{d x^2} + x \left( \frac{dy}{dx} \right)^2 = 2 x^2 \log \left( \frac{d^2 y}{d x^2} \right)\]
Write the order of the differential equation of the family of circles touching X-axis at the origin.
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 order of the differential equation whose general solution is given by y = c1 cos (2x + c2) − (c3 + c4) ax + c5 + c6 sin (x − c7) is
Write the order and the degree of the following differential equation: `"x"^3 ((d^2"y")/(d"x"^2))^2 + "x" ((d"y")/(d"x"))^4 = 0`
Determine the order and degree of the following differential equation:
`(("d"^2"y")/"dx"^2)^2 + cos ("dy"/"dx") = 0`
Determine the order and degree of the following differential equation:
`(("d"^3"y")/"dx"^3)^(1/2) - ("dy"/"dx")^(1/3) = 20`
Determine the order and degree of the following differential equation:
`"dy"/"dx" = 3"y" + root(4)(1 + 5 ("dy"/"dx")^2)`
Choose the correct alternative.
The order and degree of `[ 1+ (dy/dx)^3]^(2/3) = 8 (d^3y)/dx^3` are respectively.
Find the order and degree of the following differential equation:
`[ (d^3y)/dx^3 + x]^(3/2) = (d^2y)/dx^2`
The order and degree of `((dy)/(dx))^3 - (d^3y)/(dx^3) + ye^x` = 0 are ______.
Order of highest derivative occurring in the differential equation is called the ______ of the differential equation
Order and degree of differential equation are always ______ integers
State whether the following statement is True or False:
The degree of a differential equation `"e"^(-("d"y)/("d"x)) = ("d"y)/("d"x) + "c"` is not defined
Degree of the given differential equation
`(("d"^3"y")/"dx"^2)^2 = (1 + "dy"/"dx")^(1/3)` is
The order of the differential equation of all circles whose radius is 4, is ______.
If m and n are the order and degree of the differential equation `((d^3y)/(dx^3))^6+5((d^3y)/(dx^3))^4/((d^4y)/(dx^4))+(d^4y)/(dx^4)=x^3-1,` then ______.
The order of the differential equation of all circles of radius r, having centre on X-axis and passing through the origin is ______.
The order and degree of the differential equation `[1 + ("dy"/"dx")^2]^2 = ("d"^2y)/("d"x^2)` respectively, are ______.
The order of the differential equation of all circles of given radius a is ______.
The degree of the differential equation `("d"^2y)/("d"x^2) + (("d"y)/("d"x))^3 + 6y^5` = 0 is ______.
The degree of differential equation `((d^2y)/(dx^2))^3 + ((dy)/(dx))^2 + sin((dy)/(dx)) + 1` = 0 is:
y2 = (x + c)3 is the general solution of the differential equation ______.
The degree of the differential equation `dy/dx - x = (y - x dy/dx)^-4` is ______.
The degree of the differential equation `((d^2y)/dx^2)^2 + (dy/dx)^3` = ax is 3.
The order and the degree of the differential equation `(1 + 3 dy/dx)^2 = 4 (d^3y)/(dx^3)` respectively are ______.
The degree of the differential equation `[1 + (dy/dx)^2]^3 = ((d^2y)/(dx^2))^2` is ______.
For the differential equation \[xy \frac{d^{2}y}{dx^{2}} + x \left( \frac{dy}{dx} \right)^{2} - y \frac{dy}{dx} = 0\], what is the order?
For the differential equation \[xy \frac{d^{2}y}{dx^{2}} + x \left( \frac{dy}{dx} \right)^{2} - y \frac{dy}{dx} = 0\], what is the degree?
For the differential equation \[y''' + y^{2} + e^{y'} = 0\], is the degree defined?
Based on the differential equation \[\left(\frac{d^{3}y}{dx^{3}}\right)^{2} + 5\frac{dy}{dx} = \sin(x)\], what is the degree?
