Advertisements
Advertisements
प्रश्न
Show that the given differential equation is homogeneous and solve them.
`{xcos(y/x) + ysin(y/x)}ydx = {ysin (y/x) - xcos(y/x)}xdy`
Advertisements
उत्तर
Given differential equation
`{x cos (y/x) + y sin (y/x)} y dx = {y sin (y/x) - x cos (y/x)} x dy`
or `dy/dx = ({x cos (y/x) + y sin (y/x)} y)/({y sin (y/x) - x cos (y/x)} x)`
and `dy/dx = ((y/x) {cos (y/x) + y/x sin (y/x)})/({y/x sin (y/x) - cos (y/x)} x) = g (y/x)` (say) .... (i)
The right side of the differential equation is in the form of `g (y/x)`. Therefore, this is an even exponential differential equation of zero degree.
∴ Putting y = vx
v + x `(dv)/dx = ((cos v + v sin v) v)/(v sin v - cos v)`
⇒ x `(dv)/dx = (v cos v + v^2 sin v)/(v sin v - cos v) - v`
= v cos v + v2 sin v
⇒ x `(dv)/dx = (- v^2 sin v + v cos v)/(v sin v - cos v)`
⇒ x `(dv)/dx = (2v cos v)/(v sin v - cos v)`
`= (v sin v - cos v)/(v cos v) dv = 2/x dx`
`= (tan v - 1/v) dv = 1/x dx`
On integrating
log sec v - log v = 2 log x + log C
log `((sec v)/v)` = log x2 = log C
log `((sec v)/v)` = log cx2
sec v = v. Cx2
Finally, on putting `y/x` in place of v
`sec (y/x) = (y/x). Cx^2`
`sec (y/x) = Cxy`
`xy cos |y/x| = C`
APPEARS IN
संबंधित प्रश्न
Show that the differential equation 2yx/y dx + (y − 2x ex/y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.
Find the particular solution of the differential equation:
2y ex/y dx + (y - 2x ex/y) dy = 0 given that x = 0 when y = 1.
Show that the given differential equation is homogeneous and solve them.
`y' = (x + y)/x`
Show that the given differential equation is homogeneous and solve them.
(x – y) dy – (x + y) dx = 0
Show that the given differential equation is homogeneous and solve them.
`x dy - y dx = sqrt(x^2 + y^2) dx`
Show that the given differential equation is homogeneous and solve them.
`x dy/dx - y + x sin (y/x) = 0`
Show that the given differential equation is homogeneous and solve them.
`y dx + x log(y/x)dy - 2x dy = 0`
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
Prove that x2 – y2 = c(x2 + y2)2 is the general solution of the differential equation (x3 – 3xy2)dx = (y3 – 3x2y)dy, where C is parameter
(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0
(x2 + 3xy + y2) dx − x2 dy = 0
Solve the following initial value problem:
\[\frac{dy}{dx} = \frac{y\left( x + 2y \right)}{x\left( 2x + y \right)}, y\left( 1 \right) = 2\]
Find the particular solution of the differential equation x cos\[\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\], given that when x = 1, \[y = \frac{\pi}{4}\]
Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]
A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution
Solve the following differential equation:
`(1 + 2"e"^("x"/"y")) + 2"e"^("x"/"y")(1 - "x"/"y") "dy"/"dx" = 0`
Solve the following differential equation:
`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`
Solve the following differential equation:
`x^2 dy/dx = x^2 + xy + y^2`
Solve the following differential equation:
(9x + 5y) dy + (15x + 11y)dx = 0
State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2) "d"x` and solve it
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.
Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`
A homogeneous differential equation of the `(dx)/(dy) = h(x/y)` can be solved by making the substitution.
If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x2dy = (2xy + y2)dx, then `f(1/2)` is equal to ______.
The differential equation y' = `y/(x + sqrt(xy))` has general solution given by:
(where C is a constant of integration)
