Advertisements
Advertisements
Question
The differential equation `y dy/dx + x = 0` represents family of ______.
Options
circles
parabolas
ellipses
hyperbolas
Advertisements
Solution
The differential equation `y dy/dx + x = 0` represents family of circles.
Explanation:
`y dy/dx + x = 0`
∴ `int y dy + int x dx = C`
∴ `y^2/2 +x^2/2 = C`
∴ x2 + y2 = 2C, which is a circle.
RELATED QUESTIONS
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
Verify that y = cx + 2c2 is a solution of the differential equation
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
xy dy = (y − 1) (x + 1) dx
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
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.
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 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?
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.
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
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.
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| xy = log y + k | y' (1 - xy) = y2 |
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
Solve the following differential equation.
`dy/dx = x^2 y + y`
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution is
Solve:
(x + y) dy = a2 dx
y2 dx + (xy + x2)dy = 0
`xy dy/dx = x^2 + 2y^2`
Solve the differential equation `("d"y)/("d"x) + y` = e−x
Solve: `("d"y)/("d"x) + 2/xy` = x2
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
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
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha - y cos alpha` = 0, then the value of `a^2 + b^2` is
A man is moving away from a tower 41.6 m high at a rate of 2 m/s. If the eye level of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 m from the foot of the tower, is
