- #1
- 1,752
- 143
[tex]\begin{array}{l}
\frac{{dy}}{{dx}} = \frac{{4x^2 + 5xy + y^2 }}{{x^2 }} \\
\\
\frac{{dy}}{{dx}} = 4 + \frac{{5y}}{x} + \left( {\frac{y}{x}} \right)^2 \\
\\
{\rm{Let }}v = y/x\,\,\,\, \Rightarrow \,\,\,\,y = vx \\
\\
\frac{{dy}}{{dx}} = 4 + \frac{{5v}}{{xx}} + \left( {\frac{{vx}}{x}} \right)^2 \\
\\
\frac{{dy}}{{dx}} = 4 + \frac{{5v}}{{x^2 }} + v^2 \\
\end{array}[/tex]
But the class notes say the final line should be
[tex]\frac{dy}{dx}=x\frac{dv}{dx}+v[/tex]
How did he get that?
\frac{{dy}}{{dx}} = \frac{{4x^2 + 5xy + y^2 }}{{x^2 }} \\
\\
\frac{{dy}}{{dx}} = 4 + \frac{{5y}}{x} + \left( {\frac{y}{x}} \right)^2 \\
\\
{\rm{Let }}v = y/x\,\,\,\, \Rightarrow \,\,\,\,y = vx \\
\\
\frac{{dy}}{{dx}} = 4 + \frac{{5v}}{{xx}} + \left( {\frac{{vx}}{x}} \right)^2 \\
\\
\frac{{dy}}{{dx}} = 4 + \frac{{5v}}{{x^2 }} + v^2 \\
\end{array}[/tex]
But the class notes say the final line should be
[tex]\frac{dy}{dx}=x\frac{dv}{dx}+v[/tex]
How did he get that?