Advertisements
Advertisements
प्रश्न
Find the general solution of the following differential equation:
`x (dy)/(dx) = y - xsin(y/x)`
Advertisements
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
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
Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
The general solution of the differential equation `(y dx - x dy)/y = 0` 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 x-axis.
Form the differential equation having \[y = \left( \sin^{- 1} x \right)^2 + A \cos^{- 1} x + B\], where A and B are arbitrary constants, as its general solution.
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 xy = a ex + b e−x + x2 is a solution of the differential equation \[x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0.\]
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.
From x2 + y2 + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.
\[\frac{dy}{dx} = \sin^3 x \cos^4 x + x\sqrt{x + 1}\]
\[\frac{dy}{dx} = y^2 + 2y + 2\]
\[\frac{dy}{dx} = x^2 e^x\]
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]
cosec x (log y) dy + x2y dx = 0
Solve the differential equation:
cosec3 x dy − cosec y dx = 0
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 `x^xdy + (ye^x + 2x) dx` = 0
The general solution of the differential equation y dx – x dy = 0 is ______.
Solve the differential equation: y dx + (x – y2)dy = 0
