Advertisements
Advertisements
Question
Find the equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`
Advertisements
Solution
We have,
\[\frac{dy}{dx} = \frac{2x}{y^2}\]
\[ \Rightarrow y^2 dy = 2x dx\]
Integrating both sides, we get
\[\int y^2 dy = 2\int x dx\]
\[ \Rightarrow \frac{y^3}{3} = x^2 + C . . . . . \left( 1 \right)\]
Now the given curve passes theough (- 2, 3)
Therefore, when x = - 2, y = 3
Substituting x = - 2 and y = 3 in (1) we get
\[\frac{3^3}{3} = \left( - 2 \right)^2 + C\]
\[ \Rightarrow 9 = 4 + C\]
\[ \Rightarrow C = 5\]
Putting the value of `C` in (1), we get
\[\frac{y^3}{3} = x^2 + 5\]
\[ \Rightarrow y^3 = 3 x^2 + 15\]
\[ \Rightarrow y = \left( 3 x^2 + 15 \right)^\frac{1}{3}\]
APPEARS IN
RELATED QUESTIONS
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
The solution of the differential equation (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
Which of the following differential equations has y = x as one of its particular solution?
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sin^{- 1} x\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x2 + 1) dx, x ≠ 0.
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
Find the differential equation of all non-horizontal lines in a plane.
Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Solution of differential equation xdy – ydx = 0 represents : ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
Find the general solution of the differential equation:
`log((dy)/(dx)) = ax + by`.
