Advertisements
Advertisements
Question
The slope of the tangent at a point P (x, y) on a curve is \[\frac{- x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.
Advertisements
Solution
According to the question,
\[\frac{dy}{dx} = \frac{- x}{y}\]
\[ \Rightarrow y dy = - x dx \]
ntegrating both sides with respect to x, we get
\[\int y dy = - \int x dx\]
\[ \Rightarrow \frac{y^2}{2} = - \frac{x^2}{2} + C\]
\[\text{ Since the curve passes through }\left( 3, - 4 \right),\text{ it satisfies the above equation . }\]
\[ \therefore \frac{\left( - 4 \right)^2}{2} = - \frac{3^2}{2} + C\]
\[ \Rightarrow 8 = - \frac{9}{2} + C\]
\[ \Rightarrow C = \frac{25}{2}\]
Putting the value of C, we get
\[\frac{y^2}{2} = - \frac{x^2}{2} + \frac{25}{2}\]
\[ \Rightarrow x^2 + y^2 = 25\]
APPEARS IN
RELATED QUESTIONS
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
x cos2 y dx = y cos2 x dy
tan y dx + sec2 y tan x dy = 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.
y ex/y dx = (xex/y + y) dy
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
Solve the following initial value problem:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
A population grows at the rate of 5% per year. How long does it take for the population to double?
The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
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.
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
Solve the differential equation:
`"x"("dy")/("dx")+"y"=3"x"^2-2`
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
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.
`dy/dx = x^2 y + y`
Solve the following differential equation.
`dy/dx + y` = 3
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution is
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
`xy dy/dx = x^2 + 2y^2`
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0
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?
