Advertisements
Advertisements
प्रश्न
Find the general solution of the following differential equation:
`x (dy)/(dx) = y - xsin(y/x)`
उत्तर
We have the differential equation:
`(dy)/(dx) = y/x - sin(y/x)`
The equation is a homogeneous differential equation.
Putting `y = vx ⇒ (dy)/(dx) = v + x (dv)/(dx)`
The differential equation becomes
`v + x (dv)/(dx) = v - sinv`
⇒ `(dv)/(sinv) = - (dx)/x` ⇒ cosecvdv = `-(dx)/x`
Integrating both sides, we get
`log|"cosec"v - cotv| = - log|x| + logK, K > 0` (Here, log K is an arbitrary constant.)
⇒ `log|("cosec"v - cotv)x| = log K`
⇒ `|("cosec"v - cotv)x| = K`
⇒ `("cosec"v - cotv)x = +- K`
⇒ `("cosec" y/x - cot y/x)x = C`, which is the required general solution.
APPEARS IN
संबंधित प्रश्न
Find the differential equation of the family of lines passing through the origin.
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y2 = a (b2 – x2)
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = a e3x + b e– 2x
The general solution of the differential equation `(y dx - x dy)/y = 0` is ______.
The general solution of a differential equation of the type `dx/dy + P_1 x = Q_1` is ______.
Find the differential equation representing the family of curves `y = ae^(bx + 5)`. where a and b are arbitrary constants.
Find the differential equation of all the circles which pass through the origin and whose centres lie on y-axis.
Find the differential equation of all the circles which pass through the origin and whose centres lie on x-axis.
The equation of the curve satisfying the differential equation y (x + y3) dx = x (y3 − x) dy and passing through the point (1, 1) is
Verify that y = A cos x + sin x satisfies the differential equation \[\cos x\frac{dy}{dx} + \left( \sin x \right)y=1.\]
Find the differential equation corresponding to y = ae2x + be−3x + cex where a, b, c are arbitrary constants.
\[\frac{dy}{dx} = x^2 e^x\]
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]
\[(\tan^2 x + 2\tan x + 5)\frac{dy}{dx} = 2(1+\tan x)\sec^2x\]
(1 + x) y dx + (1 + y) x dy = 0
cos y log (sec x + tan x) dx = cos x log (sec y + tan y) dy
(1 − x2) dy + xy dx = xy2 dx
If n is any integer, then the general solution of the equation `cos x - sin x = 1/sqrt(2)` is
General solution of tan 5θ = cot 2θ is
Solution of the equation 3 tan(θ – 15) = tan(θ + 15) is
Which of the following equations has `y = c_1e^x + c_2e^-x` as the general solution?
The general solution of the differential equation `(dy)/(dx) = e^(x + y)` is
The general solution of the differential equation `(ydx - xdy)/y` = 0
The general solution of the differential equation `x^xdy + (ye^x + 2x) dx` = 0
Find the general solution of differential equation `(dy)/(dx) = (1 - cosx)/(1 + cosx)`
Solve the differential equation: y dx + (x – y2)dy = 0
The general solution of the differential equation ydx – xdy = 0; (Given x, y > 0), is of the form
(Where 'c' is an arbitrary positive constant of integration)
