Advertisements
Advertisements
Question
Show that the given differential equation is homogeneous and solve them.
(x2 – y2) dx + 2xy dy = 0
Advertisements
Solution
(x2 - y2) dx + 2xy dy = 0
Which can be written as
`dy/dx = (y^2 - x^2)/(2 xy)`
`= ((y/x)^2 - 1)/(2 (y/x))` ....(1)
Since R.H.S is of the form `g(y/x)`, and so it is a homogeneous function of degree zero
Therefore equation (1) is a homogeneous differential equation.
⇒ `dy/dx = v + x (dv)/dx`, then (1) become
`v + x (dv)/dx = (v^2 - 1)/(2v)`
⇒ `x (dv)/dx = (v^2 - 1)/(2v) - v`
⇒ `(2vdv)/(v^2 + 1) = -dx/x` ....(2)
Integrating (2) both sides, we get
log |v2 + 1| = - log |x| + C
⇒ log |(v2 + 1) x | = C
⇒ `log |(y^2 + x^2)/x| = C_1` ...`(∵ v = y/x)`
⇒ `|(y^2 + x^2)/x| = e^(C_(1))`
⇒ `(x^2 + y^2)/x =pm e^(C_(1)) = C` (say)
⇒ `x^2 + y^2 = Cx`
which is the required general solution of the given differential equation.
APPEARS IN
RELATED QUESTIONS
Solve the differential equation (x2 + y2)dx- 2xydy = 0
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 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:
`[xsin^2(y/x - y)] dx + x dy = 0; y = pi/4 "when" x = 1`
Solve the following initial value problem:
(x2 + y2) dx = 2xy dy, y (1) = 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
Which of the following is a homogeneous differential equation?
Solve the differential equation: ` (dy)/(dx) = (x + y )/ (x - y )`
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:
`x * dy/dx - y + x * sin(y/x) = 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
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`.
State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2) "d"x` and solve it
Which of the following is not a homogeneous function of x and y.
F(x, y) = `(ycos(y/x) + x)/(xcos(y/x))` is not a homogeneous function.
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
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)
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 ______.
A function \[F(x,y)\] is homogeneous of degree \[n\] when which condition holds?
After using \[y=vx\] and writing the right-hand side as \[g(v)\], which separable form is obtained?
Which substitutions are chosen for a homogeneous differential equation?
Which sequence correctly states the essential actions for obtaining the final answer of a homogeneous differential equation?
