Advertisements
Advertisements
Question
Solve the differential equation :
`y+x dy/dx=x−y dy/dx`
Advertisements
Solution
`y+x dy/dx=x−y dy/dx`
`x dy/dx + y dy/dx=x−y`
`⇒dy/dx=(x−y)/(x+y) ` ......(1)
`Let F(x, y) =(x−y)/(x+y)`
`F(λx, λy) = λF(x, y)`
Therefore, F(x, y) is a homogeneous function of degree zero.
Let `y=vx`
`dy/dx=v+x (dv)/dx`
Substituting the value of y and dy/dx in (1) we get,
`v + x (dv)/dx=(x−vx)/(x+vx)=(1−v)/(1+v)`
`x (dv)/dx=(1−v)/(1+v)−v=(1−v−v^2−v)/(1+v)=(1−2v−v^2)/(1+v)`
`(1+v)/(v^2+2v−1)dv=−dx/x`
Integrating both sides, we have
`1/2 log∣(y^2/x^2)+(2y)/x−1∣+log|x|=logc`
`⇒log∣(y^2/x^2)+(2y)/x−1∣+2log|x|=2logc`
`⇒log((y^2/x^2)+(2y)/x−1)(x^2)=logc^2`
`⇒((y^2+2yx−x^2)/x^2)(x^2) = c^2`
`⇒y^2+2yx−x^2=C (where C=c^2)`
APPEARS IN
RELATED QUESTIONS
Solve the differential equation (x2 + y2)dx- 2xydy = 0
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 - y dx = sqrt(x^2 + y^2) dx`
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`
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
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 differential equation:
`x * dy/dx - y + x * sin(y/x) = 0`
Solve the following differential equation:
`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`
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:
`x^2. dy/dx = x^2 + xy + y^2`
Which of the following is not a homogeneous function of x and y.
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.
Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`
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 ______.
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)
