Advertisements
Advertisements
प्रश्न
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
Advertisements
उत्तर
y = x n
Differentiating w.r.t. x, we get
`dy/dx = nx^(n-1)`
Again, differentiating w.r.t. x, we get
`(d^2y)/dx^2 = n(n-1) x^(n-2)`
∴ `x^2(d^2y)/dx^2 - nxdy/dx +ny`
= n(n-1)x2xn-2 - nx.nxn-1+ nxn
= n(n-1)xn - n2 xn + nxn
=[n(n-1)-n2+n]xn
= 0
∴ `x^2 (d^2y)/dx^2 - nxdy/dx + ny = 0`
∴ Given function is a solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
tan y dx + sec2 y tan x dy = 0
(y2 + 1) dx − (x2 + 1) dy = 0
(y2 − 2xy) dx = (x2 − 2xy) dy
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:-
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]
Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\] = x (x + 1) and passing through (1, 0).
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
For each of the following differential equations find the particular solution.
`y (1 + logx)dx/dy - x log x = 0`,
when x=e, y = e2.
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Solve the following differential equation.
dr + (2r)dθ= 8dθ
The integrating factor of the differential equation `dy/dx - y = x` is e−x.
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve: `("d"y)/("d"x) + 2/xy` = x2
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
Solve the differential equation `"dy"/"dx" + 2xy` = y
Which of the following is an example of an ordinary differential equation?
