Understanding ODE Substitution with a Practical Example

[tony873004]

This is another example from the blackboard that I'm trying to understand.

$$\frac{{dy}}{{dx}} = \frac{{x^2 + 3xy + y^2 }}{{x^2 }}$$

Divide through by x^2
$$\frac{{dy}}{{dx}} = 1 + \frac{{3y}}{x} + \left( {\frac{y}{x}} \right)^2$$

make substitution
$$let\,\,v = \frac{y}{x}$$

Therefore,
$$y = vx\,$$

But the next step in the example says
$$y = vx\frac{{dy}}{{dx}} = v\frac{{dv}}{{dx}}$$

How did the dy/dx pop in there?

 

[Dick]

Your 'copying from the blackboard' is wildly inaccurate. Why don't you just try to solve it without trying to verify inaccurate notes? If y=v*x, dy/dx=x*dv/dx+v.

 

[tony873004]

...If y=v*x, dy/dx=x*dv/dx+v.

Thanks, Dick. That got me through the rest of the problem.