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.
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.
`x dy - y dx = sqrt(x^2 + y^2) dx`
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`
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:
x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Which of the following is a homogeneous differential equation?
(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:
(xy − y2) dx − x2 dy = 0, y(1) = 1
Solve the following initial value problem:
(y4 − 2x3 y) dx + (x4 − 2xy3) dy = 0, y (1) = 1
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]
Which of the following is a homogeneous differential equation?
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 differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.
Solve the following differential equation:
(x2 – y2)dx + 2xy dy = 0
Which of the following is not a homogeneous function of x and y.
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.
F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
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)
For the substitution \[x=vy\], which differentiated form is correct?
Which dependence is checked on the right-hand side of a homogeneous differential equation?
Which substitutions are chosen for a homogeneous differential equation?
Which separable equation follows from \[x\frac{dv}{dx}=\frac{1}{\cos v}\]?
