Advertisements
Advertisements
Question
Show that the given differential equation is homogeneous and solve them.
(x – y) dy – (x + y) dx = 0
Advertisements
Solution
(x - y) dy - (x + y) dx = 0
`=> dy/dx = (x + y)/(x - y)`
`= (1 + (y/x))/(1 - (y/x))`
∵ The powers of the numerator and denominator are the same so this is a homogeneous differential equation.
∴ Putting y = vx
From equation (i),
`dy/dx = v + x (dv)/dx`
`v + x (dv)/dx = (x + vx)/(x - vx)`
`=> x (dv)/dx = (1 + v)/(1 - v) - v`
`=> x dy/dx= (1 + v - v + v^2)/(1 - v)`
`=> x (dv)/dx = (1 + v^2)/(1 - v)`
`=> ((1 - v)/(1 + v^2)) dv = dx/x`
Integrating on both sides
`=> int ((1 - v)/(1 + v^2)) dv = 1/x dx`
`=> int 1/(v^2 + 1) v - 1/2 int (2v)/(v^2 + 1) dv = int 1/x dx`
`=> tan^-1 v = 1/2 log (v^2 + 1) + log x + C`
`=> tan^-1 v = 1/2 log (v^2 + 1) + log x + C`
`=> tan^-1 (y/x) = 1/2 log (y^2/x^2 + 1) + log x + C because y = vx`
`=> tan^-1 (y/x) = 1/2 log ((y^2 + x^2)/x^2) + log x + C`
`=> tan^-1 (y/x) = 1/2 log (x^2 + y^2) - 1/2 log x^2 + log x + C`
`=> tan^-1 (y/x) = 1/2 log (x^2 + y^2) + C`
APPEARS IN
RELATED QUESTIONS
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 differential equation `2xydy/dx=x^2+3y^2` is homogeneous and solve it.
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.
`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:
x2 dy + (xy + y2) dx = 0; y = 1 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
(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:
\[x e^{y/x} - y + x\frac{dy}{dx} = 0, y\left( e \right) = 0\]
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:
\[\frac{dy}{dx} = \frac{y\left( x + 2y \right)}{x\left( 2x + y \right)}, y\left( 1 \right) = 2\]
Solve the following initial value problem:
(y4 − 2x3 y) dx + (x4 − 2xy3) dy = 0, y (1) = 1
Solve the following initial value problem:
x (x2 + 3y2) dx + y (y2 + 3x2) dy = 0, y (1) = 1
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]
A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution
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:
y2 dx + (xy + x2)dy = 0
Solve the following differential equation:
`"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 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
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) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`
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)
