Advertisements
Advertisements
प्रश्न
Show that the differential equation `2xydy/dx=x^2+3y^2` is homogeneous and solve it.
Advertisements
उत्तर
The given differential equation can be expressed as
`dy/dx=(x^2+3y^2)/(2xy) .....(i)`
`Let F(x, y)=(x^2+3y^2)/(2xy)`
Now,
`F(λx, λy)=((λx)^2+3(λy)^2)/(2(λx)(λy))=(λ^2(x^2+3y^2))/(λ^2(2xy))=λ^0F(x, y)`
Therefore, F(x, y) is a homogenous function of degree zero. So, the given differential equation is a homogenous differential equation.
Let y = vx .....(ii)
Differentiating (ii) w.r.t. x, we get
`dy/dx=v+x(dv)/dx`
Substituting the value of y and dy/dx in (i), we get
`v+x(dv)/dx=(1+3v^2)/(2v)`
`⇒x(dv)/dx=(1+3v^2)/(2v)−v`
` ⇒x(dv)/dx=(1+3v^2−2v^2)/(2v)`
`⇒x(dv)/dx=(1+v^2)/(2v)`
`⇒(2v)/(1+v^2)dv=dx/x .....(ii)`
Integrating both side of (iii), we get
`∫(2v)/(1+v^2)dv=∫dx/x`
Putting `1+v^2=t`
⇒2vdv=dt
`∴∫dt/t=∫dx/x`
⇒log|t|=log|x| +log|C1|
⇒log∣t/x∣=log|C1|
`⇒t/x=±C_1`
`⇒(1+v^2)/x=±C_1`
`⇒(1+y^2/x^2)/x=±C_1`
x2+y2=Cx3
APPEARS IN
संबंधित प्रश्न
Show that the given differential equation is homogeneous and solve them.
`y' = (x + y)/x`
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`
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
For the differential equation find a particular solution satisfying the given condition:
`2xy + y^2 - 2x^2 dy/dx = 0; y = 2` when x = 1
A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.
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.
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:
\[x e^{y/x} - y + x\frac{dy}{dx} = 0, y\left( e \right) = 0\]
Solve the following initial value problem:
x (x2 + 3y2) dx + y (y2 + 3x2) dy = 0, y (1) = 1
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]
Solve the following differential equation:
`"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0`
State whether the following statement is True or False:
A homogeneous differential equation is solved by substituting y = vx and integrating it
Find the equation of a curve passing through `(1, pi/4)` if the slope of the tangent to the curve at any point P(x, y) is `y/x - cos^2 y/x`.
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.
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)`
