Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`
Advertisements
उत्तर
`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`
∴ `"x"^2 "dy"/"dx" + "xy""dy"/"dx" = "y"^2`
∴ `("x"^2 + "xy")"dy"/"dx" = "y"^2`
∴ `"dy"/"dx" = "y"^2/("x"^2 + "xy")` ......(1)
Put y = vx
∴ `"dy"/"dx" = "v + x" "dv"/"dx"`
∴ (1) becomes, `"v + x""dv"/"dx" = ("v"^2"x"^2)/("x"^2 + "x"*"vx") = "v"^2/(1 + "v")`
∴ `"x""dv"/"dx" = "v"^2/(1 + "v") - "v" = ("v"^2 - "v" - "v"^2)/(1 + "v")`
∴ `"x""dv"/"dx" = (- "v")/(1 + "v")`
∴ `(1 + "v")/"v" "dv" = - 1/"x" "dx"`
Integrating, we get
`int (1 + "v")/"v" "dv" = - int 1/"x" "dx"`
`int (1/"v" + 1)"dv" = - int1/"x" "dx"`
∴ `int 1/"v" "dv" + int 1 "dv" = - int 1/"x" "dx"`
∴ log |v| + v = - log |x| + c
∴ log `|"y"/"x"| + "y"/"x" = - log |"x"| + "c"`
∴ log |y| - log |x| + `"y"/"x"` = - log |x| + c
∴ `"y"/"x" + log |"y"| = "c"`
This is the general solution.
APPEARS IN
संबंधित प्रश्न
Solve the differential equation :
`y+x dy/dx=x−y dy/dx`
Show that the differential equation `2xydy/dx=x^2+3y^2` is homogeneous and solve it.
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.
`{xcos(y/x) + ysin(y/x)}ydx = {ysin (y/x) - xcos(y/x)}xdy`
Show that the given differential equation is homogeneous and solve them.
`x dy/dx - y + x sin (y/x) = 0`
For the differential equation find a particular solution satisfying the given condition:
x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
For the differential equation find a particular solution satisfying the given condition:
`[xsin^2(y/x - y)] dx + x dy = 0; y = pi/4 "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
(2x2 y + y3) dx + (xy2 − 3x3) dy = 0
Solve the following initial value problem:
\[x e^{y/x} - y + x\frac{dy}{dx} = 0, y\left( e \right) = 0\]
Solve the following initial value problem:
\[\frac{dy}{dx} - \frac{y}{x} + cosec\frac{y}{x} = 0, y\left( 1 \right) = 0\]
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
Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]
Find the particular solution of the differential equation \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.
Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]
Solve the differential equation: ` (dy)/(dx) = (x + y )/ (x - y )`
Solve the following differential equation:
`"x" sin ("y"/"x") "dy" = ["y" sin ("y"/"x") - "x"] "dx"`
Solve the following differential equation:
y2 dx + (xy + x2)dy = 0
Solve the following differential equation:
`x * dy/dx - y + x * sin(y/x) = 0`
Solve the following differential equation:
`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 0`
Solve the following differential equation:
`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`
Solve the following differential equation:
(9x + 5y) dy + (15x + 11y)dx = 0
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
Find the equation of a curve passing through `(1, pi/4)` if the slope of the tangent to the curve at any point P(x, y) is `y/x - cos^2 y/x`.
F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
The solution of the differential equation `(1 + e^(x/y)) dx + e^(x/y) (1 + x/y) dy` = 0 is
A homogeneous differential equation of the `(dx)/(dy) = h(x/y)` can be solved by making the substitution.
Let the solution curve of the differential equation `x (dy)/(dx) - y = sqrt(y^2 + 16x^2)`, y(1) = 3 be y = y(x). Then y(2) is equal to ______.
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)
