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Show that the differential equation 2xy dy/dx=x^2+3y^2 is homogeneous and solve it. - Mathematics

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Show that the differential  equation `2xydy/dx=x^2+3y^2`  is homogeneous and solve it.

 
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Solution

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|

logt/x=log|C1|

`⇒t/x=±C_1`

`⇒(1+v^2)/x=±C_1`

`⇒(1+y^2/x^2)/x=±C_1`

 x2+y2=Cx3

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
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