Advertisements
Advertisements
प्रश्न
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Advertisements
उत्तर
The equation of the parabola having vertex at origin and axis along the positive direction of y-axis is given by
x2 =4ay .....(1)
Since there is only one parameter, so we differentiate it only once.
Differentiating with respect to x, we get
\[2x = 4ay'\]
\[ \Rightarrow 4a = \frac{2x}{y'}\]
Substituting the value of 4a in (1), we get
\[x^2 = \frac{2x}{y'} \times y\]
\[ \Rightarrow xy' = 2y\]
\[ \Rightarrow xy' - 2y = 0\]
APPEARS IN
संबंधित प्रश्न
If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = ex + e−x
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
3x2 dy = (3xy + y2) dx
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
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.
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The solution of the differential equation y1 y3 = y22 is
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
The price of six different commodities for years 2009 and year 2011 are as follows:
| Commodities | A | B | C | D | E | F |
|
Price in 2009 (₹) |
35 | 80 | 25 | 30 | 80 | x |
| Price in 2011 (₹) | 50 | y | 45 | 70 | 120 | 105 |
The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution is
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 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`
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
