Advertisements
Advertisements
Question
Show that the given differential equation is homogeneous and solve them.
`y dx + x log(y/x)dy - 2x dy = 0`
Advertisements
Solution
`y dx + x log (y/x) dy - 2x dy = 0`
⇒ `y/x + log (y/x) dy/dx - 2 dy/dx = 0`
⇒ `dy/dx = (y/x)/(2 - log (y/x))` .....(1)
Since R.H.S. is of the form `g(y/x)`, 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,` then (1) becomes
`v + x (dv)/dx = v/ (2 - log v)`
⇒ `x (dv)/dx = v/ (2 - log v) - v`
⇒ `((2 - log v)dv)/(v log v - v) = dx/x`
⇒ `(1 - (log v - 1))/(v (log v - 1)) dv = dx/x` .....(2)
Integrating (2) both sides, we get
`int (1/ (v (logv - 1)) - 1/v) dv`
`= int dx/x + C_1`
⇒ `int (1/v)/(log v - 1) dv - int 1/v dv = int dx /x + C_1`
⇒ `log |log v - 1| - log |v|`
= log |x| + C1
⇒ `log |(log v - 1)/(v x)| = C_1`
⇒ `|(log v - 1)/ (v x)| = e^(C_(1))`
⇒ `(log v - 1)/(v x) = pm e^(C_(1)) = C` (say)
⇒ `log (y/x) - 1 = Cy` ...`(∵ v = y/x)`
which is the required general solution.
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.
(x – y) dy – (x + y) dx = 0
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.
`{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.
`(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
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`
(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 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:
(xy − y2) dx − x2 dy = 0, y(1) = 1
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
Find the particular solution of the differential equation \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.
Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]
A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution
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:
y2 dx + (xy + x2)dy = 0
Solve the following differential equation:
`"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0`
Solve the following differential equation:
`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`
Solve the following differential equation:
x dx + 2y dx = 0, when x = 2, y = 1
The solution of the differential equation `(1 + e^(x/y)) dx + e^(x/y) (1 + x/y) dy` = 0 is
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 ______.
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 differential equation y2 dx + (x2 − xy + y2)dy = 0 is ______.
After using \[y=vx\] and writing the right-hand side as \[g(v)\], which separable form is obtained?
Writing \[x\cos\left(\frac{y}{x}\right)\frac{dy}{dx}=y\cos\left(\frac{y}{x}\right)+x\] in standard form gives which expression?
From \[v+x\frac{dv}{dx}=\frac{v\cos v+1}{\cos v}\], what is \[x\frac{dv}{dx}\]?
