Advertisements
Advertisements
Question
Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2).
Advertisements
Solution
Portion of the x-axis cut off between the origin and tangent at a point \[= x - y \hspace{0.167em} \hspace{0.167em} \frac{dx}{dy} = OT\]
It is given, OT = 2x
\[\begin{array}{l}\therefore \hspace{0.167em} \hspace{0.167em} x - y \hspace{0.167em} \hspace{0.167em} \frac{dx}{dy} = 2x \\ - x = y\frac{dx}{dy} \\ - \int\frac{dx}{x} = \int\frac{dy}{y} \\ \therefore \hspace{0.167em} \hspace{0.167em} xy = k\end{array}\]
Since the curve passes through the point (1, 2)
⇒ at x = 1 ⇒ y = 2
∴ k = 2
∴ xy = 2
APPEARS IN
RELATED QUESTIONS
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
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
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] 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.
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
A population grows at the rate of 5% per year. How long does it take for the population to double?
Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e2x.
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.
Show that all curves for which the slope at any point (x, y) on it is \[\frac{x^2 + y^2}{2xy}\] are rectangular hyperbola.
Define a differential equation.
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
The solution of the differential equation y1 y3 = y22 is
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
The differential equation `y dy/dx + x = 0` represents family of ______.
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = ex | `dy/ dx= y` |
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
Solve the differential equation:
`e^(dy/dx) = x`
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
Solve the differential equation `"dy"/"dx" + 2xy` = y
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
