Can we use y=vx for non-homogeneous differential equation?

[jasoncurious]

Homework Statement

Can we use y=vx for non-homogeneous differential equation?

Example:
yy'=x^3+(y^2/x)→not homogeneous

Homework Equations

y=vx
dy/dx=v+x(dv/dx)

The Attempt at a Solution

By substituting the equation above:
vx(v+x dv/dx)=x^3+(v^2 x^2)/x
v^2*x+vx^2 dv/dx=x^3+v^2*x
Eliminate the v^2*x:
vx^2 dv/dx=x^3
Divide both sides with x^2:
v dv/dx=x
vdv=xdx
Continue the integration:
y^2=x^2(x^2+c), where c is a constant

 

[MrWarlock616]

The substitution y=vx makes it easier to find the solution, because a homogenous differential equation takes the form:

$\frac{dy}{dx}=\frac{f_1(x,y)}{f_2(x,y)}$

which then equals $\frac{f(y/x)}{g(y/x)}$ or $\frac{f(x/y)}{g(x/y)}$ by taking $x^n$ or $y^n$ common, if f₁ and f₂ are homogenous in degree n.

 

[Ray Vickson]

Homework Statement

Can we use y=vx for non-homogeneous differential equation?

Example:
yy'=x^3+(y^2/x)→not homogeneous

Homework Equations

y=vx
dy/dx=v+x(dv/dx)

The Attempt at a Solution

By substituting the equation above:
vx(v+x dv/dx)=x^3+(v^2 x^2)/x
v^2*x+vx^2 dv/dx=x^3+v^2*x
Eliminate the v^2*x:
vx^2 dv/dx=x^3
Divide both sides with x^2:
v dv/dx=x
vdv=xdx
Continue the integration:
y^2=x^2(x^2+c), where c is a constant

Easier: $v \, dv/dx = d(v^2/2)/dx.$ And, there are two solutions.

 

[jasoncurious]

May I know the hint to the other solution? Thanks