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
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.
`x dy - y dx = sqrt(x^2 + y^2) dx`
Show that the given differential equation is homogeneous and solve them.
`y dx + x log(y/x)dy - 2x dy = 0`
Show that the given differential equation is homogeneous and solve them.
`(1+e^(x/y))dx + e^(x/y) (1 - x/y)dy = 0`
For the differential equation find a particular solution satisfying the given condition:
(x + y) dy + (x – y) dx = 0; y = 1 when x = 1
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`
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
A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.
Which of the following is a homogeneous differential equation?
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.
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0
(2x2 y + y3) dx + (xy2 − 3x3) dy = 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:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
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, \[y = \frac{\pi}{4}\]
Which of the following is a homogeneous differential equation?
Solve the following differential equation:
`"x" sin ("y"/"x") "dy" = ["y" sin ("y"/"x") - "x"] "dx"`
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:
x dx + 2y dx = 0, when x = 2, y = 1
Solve the following differential equation:
(9x + 5y) dy + (15x + 11y)dx = 0
Solve the following differential equation:
(x2 + 3xy + y2)dx - x2 dy = 0
Solve the following differential equation:
(x2 – y2)dx + 2xy dy = 0
Which of the following is not a homogeneous function of x and y.
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.
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)
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
Read the following passage:
|
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form `dy/dx` = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogeneous function of degree n if F(λx, λy) = λn F(x, y). To solve a homogeneous differential equation of the type `dy/dx` = F(x, y) = `g(y/x)`, we make the substitution y = vx and then separate the variables. |
Based on the above, answer the following questions:
- Show that (x2 – y2) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
- Solve the above equation to find its general solution. (2)
The solution of the equation `dy/dx = (3x − 4y − 2)/(3x − 4y − 3)` is ______.
The solution of the differential equation y2 dx + (x2 − xy + y2)dy = 0 is ______.
