Advertisements
Advertisements
Question
Show that the given differential equation is homogeneous and solve them.
`x dy - y dx = sqrt(x^2 + y^2) dx`
Solution
`x dy - y dx = sqrt(x^2 + y^2) dx`,
Which can be written as `x dy/dx = y + sqrt (x^2 + y^2)`
or `dy/dx = y/x + sqrt (1 + (y/x)^2)` ....(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.
To solve this, put y = vx
⇒ `dy/dx = v + x (dv)/dx`
Substituting the value of y and `dy/dx` in (1), we get
`v + x (dv)/dx = v + sqrt (1 + v^2)`
⇒ `x (dv)/dx = sqrt(1 + v^2)`
⇒ `dx/x = (dv)/sqrt(1 + v^2)`
⇒ `int dx/x = int (dv)/ sqrt(1 + v^2)`
⇒ `log x + log C_1 = log |v + sqrt (1+ v^2)|`
⇒ `log x + log C_1 = log |y/x + sqrt (1 + y^2/x^2)|`
⇒ `log C_1 x = log |y + sqrt (x^2 + y^2)| - log x`
⇒ `pm C_1 x^2 = y + sqrt (x^2 + y^2)`
⇒ `Cx^2 = y + sqrt (x^2 + y^2)`
APPEARS IN
RELATED QUESTIONS
Solve the differential equation :
`y+x dy/dx=x−y dy/dx`
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.
(x2 – y2) dx + 2xy dy = 0
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.
`(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:
`[xsin^2(y/x - y)] dx + x dy = 0; y = pi/4 "when" x = 1`
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 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 + 3xy + y2) dx − x2 dy = 0
(2x2 y + y3) dx + (xy2 − 3x3) dy = 0
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:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]
Find the particular solution of the differential equation \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.
Which of the following is a homogeneous differential equation?
Solve the following differential equation:
`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`
Solve the following differential equation:
x dx + 2y dx = 0, when x = 2, y = 1
Solve the following differential equation:
(x2 – y2)dx + 2xy dy = 0
State whether the following statement is True or False:
A homogeneous differential equation is solved by substituting y = vx and integrating it
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.
Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`
The differential equation y' = `y/(x + sqrt(xy))` has general solution given by:
(where C is a constant of integration)
