Advertisements
Advertisements
प्रश्न
If `(a + bx)e^(y/x)` = x then prove that `x(d^2y)/(dx^2) = (a/(a + bx))^2`.
Advertisements
उत्तर
`y/x = log_e (x/(a + bx))` = loge x – loge (a + bx)
On differentiating with respect to x, we get
`\implies (x dy/dx - y)/x^2 = 1/x - 1/(a + bx) d/dx(a + bx) = 1/x - b/(a + bx)`
`\implies x dy/dx - y = x^2(1/x - b/(a + bx)) = (ax)/(a + bx)`
On differentiating again with respect to x, we get
`\implies x (d^2y)/(dx^2) + dy/dx - dy/dx = ((a + bx)a - ax(b))/(a + bx)^2`
`\implies x (d^2y)/(dx^2) = (a/(a + bx))^2`.
APPEARS IN
संबंधित प्रश्न
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)` = cos 3x + sin 3x
Determine the order and degree (if defined) of the differential equation:
y′ + y = ex
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
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 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
The order of the differential equation satisfying
\[\sqrt{1 - x^4} + \sqrt{1 - y^4} = a\left( x^2 - y^2 \right)\] is
Find the order and the degree of the differential equation `x^2 (d^2y)/(dx^2) = { 1 + (dy/dx)^2}^4`
Determine the order and degree of the following differential equation:
`(dy)/(dx) = (2sin x + 3)/(dy/dx)`
Choose the correct option from the given alternatives:
The order and degree of the differential equation `sqrt(1 + ("dy"/"dx")^2) = (("d"^2"y")/"dx"^2)^(3/2)` are respectively.
Determine the order and degree of the following differential equations.
`dy/dx = 7 (d^2y)/dx^2`
Determine the order and degree of the following differential equations.
`((d^3y)/dx^3)^(1/6) = 9`
State whether the following is True or False:
The order of highest derivative occurring in the differential equation is called degree of the differential equation.
Order of highest derivative occurring in the differential equation is called the ______ of the differential equation
The power of highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any is called ______ of the differential equation
Order and degree of differential equation`(("d"^3y)/("d"x^3))^(1/6)`= 9 is ______
State whether the following statement is True or False:
The degree of a differential equation is the power of highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any
Degree of the given differential equation
`(("d"^3"y")/"dx"^2)^2 = (1 + "dy"/"dx")^(1/3)` is
The degree of the differential equation `("d"^4"y")/"dx"^4 + sqrt(1 + ("dy"/"dx")^4)` = 0 is
The differential equation `x((d^2y)/dx^2)^3 + ((d^3y)/dx^3)^2y = x^2` is of ______
The third order differential equation is ______
The order of the differential equation of all circles of radius r, having centre on X-axis and passing through the origin is ______.
Degree of the differential equation `sqrt(1 + ("d"^2y)/("d"x^2)) = x + "dy"/"dx"` is not defined.
Write the degree of the differential equation (y''')2 + 3(y") + 3xy' + 5y = 0
If m and n, respectively, are the order and the degree of the differential equation `d/(dx) [((dy)/(dx))]^4` = 0, then m + n = ______.
Find the order and degree of the differential equation `(d^2y)/(dx^2) = root(3)(1 - (dy/dx)^4`
