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
Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = ex
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
(ey + 1) cos x dx + ey sin x dy = 0
Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?
Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\] and tangent at any point of which makes an angle tan−1 \[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.
Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\] = x (x + 1) and passing through (1, 0).
A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Solve the following differential equation.
`(x + y) dy/dx = 1`
For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
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`
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
