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
Determine the order and degree (if defined) of the differential equation:
y' + 5y = 0
Determine the order and degree (if defined) of the differential equation:
`((ds)/(dt))^4 + 3s (d^2s)/(dt^2) = 0`
Determine the order and degree (if defined) of the differential equation:
y′′′ + 2y″ + y′ = 0
Determine the order and degree (if defined) of the differential equation:
y″ + 2y′ + sin y = 0
For the differential equation given below, indicate its order and degree (if defined).
`(d^2y)/dx^2 + 5x(dy/dx)^2 - 6y = log x`
(xy2 + x) dx + (y − x2y) dy = 0
Define order of a differential equation.
Write the degree of the differential equation
\[a^2 \frac{d^2 y}{d x^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^{1/4}\]
Write the order of the differential equation
\[1 + \left( \frac{dy}{dx} \right)^2 = 7 \left( \frac{d^2 y}{d x^2} \right)^3\]
Write the order of the differential equation whose solution is y = a cos x + b sin x + c e−x.
Write the degree of the differential equation \[\left( \frac{dy}{dx} \right)^4 + 3x\frac{d^2 y}{d x^2} = 0\]
Write the order and degree of the differential equation
\[\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^\frac{1}{4} + x^\frac{1}{5} = 0\]
The degree of the differential equation \[\frac{d^2 y}{d x^2} + e^\frac{dy}{dx} = 0\]
Determine the order and degree (if defined) of the following differential equation:-
y" + 2y' + sin y = 0
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:
`[1 + (dy/dx)^2]^(3/2) = 8(d^2y)/dx^2`
Determine the order and degree of the following differential equations.
`(d^2x)/(dt^2)+((dx)/(dt))^2 + 8=0`
Choose the correct alternative.
The order and degree of `(dy/dx)^3 - (d^3y)/dx^3 + ye^x = 0` are respectively.
State whether the following is True or False:
The power of the highest ordered derivative when all the derivatives are made free from negative and / or fractional indices if any is called order of the differential equation.
Find the order and degree of the following differential equation:
`[ (d^3y)/dx^3 + x]^(3/2) = (d^2y)/dx^2`
State the degree of differential equation `e^((dy)/(dx)) + (dy)/(dx)` = x
Choose the correct alternative:
The order and degree of `(1 + (("d"y)/("d"x))^3)^(2/3) = 8 ("d"^3y)/("d"x^3)` are respectively
State whether the following statement is True or False:
Order and degree of differential equation `x ("d"^3y)/("d"x^3) + 6(("d"^2y)/("d"x^2))^2 + y` = 0 is (2, 2)
The order and degree of the differential equation `[1 + 1/("dy"/"dx")^2]^(5/3) = 5 ("d"^2y)/"dx"^2` are respectively.
The order of the differential equation of all circles which lie in the first quadrant and touch both the axes is ______.
The degree of the differential equation `1/2 ("d"^3"y")/"dx"^3 = {1 + (("d"^2"y")/"dx"^2)}^(5/3)` is ______.
Write the degree of the differential equation (y''')2 + 3(y") + 3xy' + 5y = 0
The differential equation representing the family of curves y2 = `2c(x + sqrt(c))`, where c is a positive parameter, is of ______.
Degree of the differential equation `sinx + cos(dy/dx)` = y2 is ______.
What is the order of the differential equation \[\frac{dy}{dx} - \cos x = 0\]?
What is the degree of the differential equation \[\frac{dy}{dx} - \cos x = 0\]?
For the differential equation \[y''' + y^{2} + e^{y'} = 0\], what is the order?
