Advertisements
Advertisements
प्रश्न
For the differential equation find a particular solution satisfying the given condition:
(x + y) dy + (x – y) dx = 0; y = 1 when x = 1
उत्तर
given (x + y) dy + (x – y) dx = 0
`=> dy/dx = (y - x)/(y + x)` ...(i)
∵ The powers of the numerator and denominator are the same so this is a homogeneous differential equation.
∴ Putting y = vx
`dy/dx = v + x (dv)/dx` ...(in equation (i))
`=> v + x (dv)/dx = (vx - x)/(vx + x)`
`=> x (dv)/dx = (v - 1)/(v + 1) - v`
`x (dv)/dx = (v - 1 - v^2 - 1)/(v + 1)`
`= - (v^2 + 1)/(v + 1)`
`(v + 1)/(v^2 + 1)dv = - 1/x dx`
On integrating,
`=> 1/2 int (2v)/(v^2 + 1)dv + int 1/(v^2 + 1) dv = - int 1/x dx`
`1/2 log (v^2 + 1) + tan^-1 (v) = - log x + C`
log (v2 + 1) + 2 tan-1 (v) = - 2 log x + 2C
So on putting `y/x` in place of v,
`log ((y^2 + x^2)/x^2) + 2 tan^-1 (y/x) = - log x^2 + 2C`
`log (x^2 + y^2) - log x^2 + 2 tan^-1 (y/x) = - log x^2 x + 2C`
`log (x^2 + y^2) + 2 tan^-1 (y/x) = 2C` ....(ii)
Given y = 1 and x = 1
log (12 + 12) + 2 tan-1 (1) = 2C
log 2 + 2 tan-1 (1) = 2C
2C = log 2 + 2 `xx pi/4 = log 2 + pi/2`
Putting this value of C in equation (ii),
`log (x^2 + y^2) + 2 tan^-1 (y/x) = pi/2 + log 2`
APPEARS IN
संबंधित प्रश्न
Solve the differential equation (x2 + y2)dx- 2xydy = 0
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.
(x2 + xy) dy = (x2 + y2) dx
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^2 dy/dx = x^2 - 2y^2 + xy`
Show that the given differential equation is homogeneous and solve them.
`y dx + x log(y/x)dy - 2x dy = 0`
For the differential equation find a particular solution satisfying the given condition:
`dy/dx - y/x + cosec (y/x) = 0; y = 0` when x = 1
For the differential equation find a particular solution satisfying the given condition:
`2xy + y^2 - 2x^2 dy/dx = 0; y = 2` when x = 1
A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.
Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3 – 3x y2) dx = (y3 – 3x2y) dy, where c is a parameter.
(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0
(2x2 y + y3) dx + (xy2 − 3x3) dy = 0
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 : \[\left[ y - x \cos\left( \frac{y}{x} \right) \right]dy + \left[ y \cos\left( \frac{y}{x} \right) - 2x \sin\left( \frac{y}{x} \right) \right]dx = 0\] .
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:
`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`
Solve the following differential equation:
(x2 + 3xy + y2)dx - x2 dy = 0
State whether the following statement is True or False:
A homogeneous differential equation is solved by substituting y = vx and integrating it
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.
F(x, y) = `(ycos(y/x) + x)/(xcos(y/x))` is not a homogeneous function.
F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
A homogeneous differential equation of the `(dx)/(dy) = h(x/y)` can be solved by making the substitution.
The differential equation y' = `y/(x + sqrt(xy))` has general solution given by:
(where C is a constant of integration)
