Advertisements
Advertisements
प्रश्न
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| ax2 + by2 = 5 | `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx` |
Advertisements
उत्तर
ax 2 + by 2 = 5
Differentiating w.r.t. x, we get
`2ax +2by dy/dx = 0` ....(i)
Again, differentiating w.r.t. x, we get
`2a + 2b(dy/dx)^2 + 2by ((d^2y)/dx^2) = 0` ........(ii)
From (i), we get
`a = - (by)/x(dy/dx)`
Substituting the value of a in (ii), we get
`- 2(by)/x(dy/dx) + 2b(dy/dx)^2 + 2by((d^2y)/dx^2) = 0`
∴`- y/x(dy/dx) + (dy/dx)^2 + y((d^2y)/dx^2) = 0`
∴`- y(dy/dx) + x(dy/dx)^2 + xy((d^2y)/dx^2) = 0`
∴`x y((d^2y)/dx^2) + x(dy/dx)^2 =y((dy)/dx) `
∴ Given function is a solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
C' (x) = 2 + 0.15 x ; C(0) = 100
(1 + x2) dy = xy dx
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
3x2 dy = (3xy + y2) dx
Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.
Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
y2 dx + (x2 − xy + y2) dy = 0
The differential equation `y dy/dx + x = 0` represents family of ______.
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Form the differential equation from the relation x2 + 4y2 = 4b2
Choose the correct alternative.
The solution of `x dy/dx = y` log y is
`dy/dx = log x`
Select and write the correct alternative from the given option for the question
Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
State whether the following statement is True or False:
The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0
y = `a + b/x`
`(dy)/(dx) = square`
`(d^2y)/(dx^2) = square`
Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`
= `x square + 2 square`
= `square`
Hence y = `a + b/x` is solution of `square`
