Advertisements
Advertisements
प्रश्न
Solve the differential equation: ` (dy)/(dx) = (x + y )/ (x - y )`
Advertisements
उत्तर
The given differential equation is:
⇒ `dy/dx = (x + y)/( x - y)` ....(1)
Let F (x, y) = `(x + y)/( x - y)`
∴ F ( λx, λy) = `(λx + λy)/( λx - λy) = (x + y)/( x - y) = λ° . F(x, y)`
Thus, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: y = vx
⇒ `d/dx (y) = d/dx (vx)`
⇒ `dy/dx = v + x (dv)/dx`
Substituting the values of y and in equation (1), we get:
`v + x (dv)/(dx) = (x + vx)/(x - vx) = (1 + v)/(1 - v)`
⇒ `x (dv)/(dx) = (1 + v)/(1 - v) - v = (1 + v - v( 1 - v))/( 1 - v)`
⇒ `x (dv)/(dx) = (1 + v^2)/(1 - v)`
⇒ `(1 - v)/(1 + v^2) (dv) = (dx)/x`
Integrating both sides, we get:
`tan^-1v - 1/2 log ( 1 + y^2 ) = log x + c`
⇒ `tan^-1 (y/x) - 1/2 log [ 1 + (y/x)^2 ] = log x + c`
⇒ `tan^-1 (y/x) - 1/2 log ((x^2 + y^2)/x^2) = log x + c`
⇒ `tan^-1 (y/x) - 1/2 [ log ((x^2 + y^2)- log x^2) ] = log x + c`
⇒ `tan^-1 (y/x) - 1/2 log (x^2 + y^2) + c`
This is the required solution of the given differential equation.
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.
Solve the differential equation :
`y+x dy/dx=x−y dy/dx`
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:
(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:
`2xy + y^2 - 2x^2 dy/dx = 0; y = 2` when x = 1
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 + 3xy + y2) dx − x2 dy = 0
Solve the following initial value problem:
(x2 + y2) dx = 2xy dy, y (1) = 0
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
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 following differential equation:
`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 0`
Solve the following differential equation:
x dx + 2y dx = 0, when x = 2, y = 1
Solve the following differential equation:
`x^2 dy/dx = x^2 + xy + y^2`
Solve the following differential equation:
(x2 + 3xy + y2)dx - x2 dy = 0
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.
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
