Advertisements
Advertisements
प्रश्न
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
उत्तर
Given equation is `x^2 "dy"/"dx"` = x2 + xy + y2
⇒ `"dy"/"dx" = (x^2 + xy + y^2)/x^2`
Put y = vx ......[∵ it is a homogeneous differential equation]
∴ `"dy"/"dx" = "v" + x * "dv"/"dx"`
∴ `"v" + x * "dv"/"dx" = (x^2 + "v"x^2 + "v"^2x^2)/x^2`
⇒ `"v" + x * "dv"/"dx" = (x^2(1 + "v" + v"^2))/x^2`
⇒ `"v" + x * "dv"/"dx" = 1 + "v" + "v"^2`
⇒ `x * "dv"/"dx" = 1 + "v" + "v"^2 - "v"`
⇒ `x * "dv"/"dx" = 1 + "v"^2`
⇒ `"dv"/(1 + "v"^2) = "dx"/x`
Integrating both sides, we get
`int "dv"/(1 + "v"^2) = int "dx"/x`
⇒ tan–1v = log x + c
⇒ `tan^-1 (y/x)` = log x + c
Hence, the required solution is `tan^-1 (y/x)` = log |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.
(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.
(x2 – y2) dx + 2xy dy = 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.
`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`
For the differential equation find a particular solution satisfying the given condition:
x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
Solve the following initial value problem:
(y4 − 2x3 y) dx + (x4 − 2xy3) dy = 0, y (1) = 1
Solve the following initial value problem:
x (x2 + 3y2) dx + y (y2 + 3x2) dy = 0, y (1) = 1
A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution
Which of the following is a homogeneous differential equation?
Solve the differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.
Solve the following differential equation:
`"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0`
Solve the following differential equation:
`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 0`
Solve the following differential equation:
`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`
Solve the following differential equation:
x dx + 2y dx = 0, when x = 2, y = 1
Solve the following differential equation:
(9x + 5y) dy + (15x + 11y)dx = 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.
The differential equation y' = `y/(x + sqrt(xy))` has general solution given by:
(where C is a constant of integration)
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
