Advertisements
Advertisements
प्रश्न
Show that the given differential equation is homogeneous and solve them.
(x2 + xy) dy = (x2 + y2) dx
Advertisements
उत्तर
Let `dy/dx = (x^2 + y^2)/(x^2 + xy) = f (xy)` ..... (i)
Now, f `(lamda x, lambda y) = (lambda^2 (x^2 + y^2))/(lambda^2 (x^2 + xy)) = lambda^0` f (x, y)
`therefore` F(x,y) is an exponential function of degree zero.
Hence the given differential equation is a homogeneous differential equation.
Now y = vx
`dy/dx = v + x (dv)/dx`
Then, from equation (i)
V + x `(dv)/dx = (x^2 + v^2 x^2)/(x^2 + vx^2)`
`=> x (dv)/dx = (1 + v^2)/(1 + v) - v`
`x (dv)/dx = (1 + v - v - v^2)/(1 + v)`
`x (dv)/dx = (dv)/dx = (1 - v)/(1 + v)`
`=> (1 + v)/(1 - v) dv = dx/x`
On integrating,
`int (1 + v)/(1 - v) dv = int 1/x dx`
`=> int (-1 + 2/(1 - v)) dv = int 1/x dx`
⇒ - v - 2 log (1 - v) = log x + log C
⇒ - v = log Cx + 2 log (1 - v)
⇒ - v = log Cx + log (1 - v)2
Cx . (1 - v)2 = e-v
On substituting `y/x` in place of v,
`C. x ((x - y)^2)/x^2 = e^(-y/x)`
`=> (x - y)^2 = Cxe^(-y/x)`
APPEARS IN
संबंधित प्रश्न
Solve the differential equation :
`y+x dy/dx=x−y dy/dx`
Find the particular solution of the differential equation:
2y ex/y dx + (y - 2x ex/y) dy = 0 given that x = 0 when y = 1.
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.
`x^2 dy/dx = x^2 - 2y^2 + xy`
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:
(x + y) dy + (x – y) dx = 0; y = 1 when x = 1
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`
Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3 – 3x y2) dx = (y3 – 3x2y) dy, where c is a parameter.
(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:
\[\frac{dy}{dx} - \frac{y}{x} + cosec\frac{y}{x} = 0, y\left( 1 \right) = 0\]
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:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]
Find the particular solution of the differential equation x cos\[\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\], given that when x = 1, \[y = \frac{\pi}{4}\]
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:
`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 0`
Solve the following differential equation:
`x^2. dy/dx = x^2 + xy + y^2`
Solve the following differential equation:
(9x + 5y) dy + (15x + 11y)dx = 0
State whether the following statement is True or False:
A homogeneous differential equation is solved by substituting y = vx and integrating 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.
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)
