Advertisements
Advertisements
प्रश्न
Solve the differential equation `ye^(x/y) dx = (xe^(x/y) + y^2)dy, (y != 0)`
Advertisements
उत्तर १
`ye^(x/y)dx = (xe^(x/y) + y^2)dy`
`\implies ye^(x/y) dx/dy = xe^(x/y) + y^2`
`\implies e^(x/y) [y.dx/dy - x]` = y2
`\implies e^(x/y). ([y.dx/dy - x])/y^2` = 1 ...(1)
Let `e^(x/y)` = z.
Differentiating it with respect to y, we get:
`d/dy(e^(x/y)) = dz/dy`
`\implies e^(x/y) . d/dy (x/y) = dz/dy`
`\implies e^(x/y). [(y.dx/dy - x)/y^2] = dz/dy` ...(2)
From equation (1) and equation (2), we get:
`dz/dy` = 1
`\implies` dz = dy
Integrating both sides, we get:
z = y + C
`\implies e^(x/y)` = y + C
उत्तर २
`ye^(x/y)dx = (xe^(x/y) + y^2)dy`
`\implies e^(x/y) (ydx - xdy)` = y2 dy
`\implies e^(x/y) ((ydx - xdy)/y^2)` = dy
`\implies e^(x/y)d(x/y)` = dy
`\implies int e^(x/y)d (x/y) = int dy`
`\implies e^(x/y)` = y + c, where 'c' is an arbitrary constant of integration.
उत्तर ३
We have, `dx/dy = (xe^(x/y) + y^2)/(y.e^(x/y)`
`\implies dx/dy = x/y + y/e^(x/y)` ...(i)
Let x = vy `\implies dx/dy = v + y.(dv)/dy`;
So equation (i) becomes `v + y (dv)/dy = v + y/e^v`
`\implies y (dv)/dy = y/e^v`
`\implies` ev dv = dy
On integrating we get,
`inte^v dv = int dy`
`\implies` ev = y + c
`\implies` ex/y = y + c
where 'c' is an arbitrary constant of integration.
संबंधित प्रश्न
Find the differential equation of the family of lines passing through the origin.
Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = a e3x + b e– 2x
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = e2x (a + bx)
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 ______.
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is ______.
Find the differential equation of all the circles which pass through the origin and whose centres lie on x-axis.
Verify that xy = a ex + b e−x + x2 is a solution of the differential equation \[x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0.\]
Show that y = C x + 2C2 is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]
Show that y2 − x2 − xy = a is a solution of the differential equation \[\left( x - 2y \right)\frac{dy}{dx} + 2x + y = 0.\]
Show that the differential equation of all parabolas which have their axes parallel to y-axis is \[\frac{d^3 y}{d x^3} = 0.\]
From x2 + y2 + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.
\[\frac{dy}{dx} = \sin^3 x \cos^4 x + x\sqrt{x + 1}\]
\[\frac{dy}{dx} = \frac{1}{x^2 + 4x + 5}\]
\[\frac{dy}{dx} = y^2 + 2y + 2\]
\[\frac{dy}{dx} = x^2 e^x\]
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]
\[(\tan^2 x + 2\tan x + 5)\frac{dy}{dx} = 2(1+\tan x)\sec^2x\]
\[\frac{dy}{dx} = \sin^3 x \cos^2 x + x e^x\]
x cos2 y dx = y cos2 x dy
cos y log (sec x + tan x) dx = cos x log (sec y + tan y) dy
Find the general solution of the differential equation `"dy"/"dx" = y/x`.
Solve the differential equation:
cosec3 x dy − cosec y dx = 0
Find the general solution of the following differential equation:
`x (dy)/(dx) = y - xsin(y/x)`
The general solution of the differential equation `(dy)/(dx) + x/y` = 0 is
The number of arbitrary constant in the general solution of a differential equation of fourth order are
The general solution of the differential equation `(ydx - xdy)/y` = 0
Find the general solution of differential equation `(dy)/(dx) = (1 - cosx)/(1 + cosx)`
The general solution of the differential equation y dx – x dy = 0 is ______.
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)
