Toronto Math Forum

\begin{equation*}
9y''+6y'+y=0
\end{equation*}
\begin{equation*}
9r^2+6r+1=0
\end{equation*}
\begin{equation*}
 (3r+1)^2=0
\end{equation*}
\begin{equation*}
r_1=r_2=-\frac{1}{3}\\
\end{equation*}
\begin{equation*}
y=C_1e^{-\frac{1}{3}t}+C_2te^{\frac{1}{3}t}
\end{equation*}

\begin{equation*}
((x+2)sin(y))+xcos(y)y'=0
\end{equation*}
\begin{equation*}
\mu=xe^x
\end{equation*}
\begin{equation*}
M_y=(x+2)cosy
\end{equation*}
\begin{equation*}
N_x=cosy
\end{equation*}
\begin{equation*}
M_y\neq N_x \Rightarrow not \ exact
\end{equation*}
\begin{equation*}
multiple \ \mu \ on \ both \ sides
\end{equation*}
\begin{equation*}
xe^x((x+2)sin(y))+x^2e^xcos(y)y'=0
\end{equation*}
\begin{equation*}
(x^2e^x+2xe^x)sin(y)+x^2e^xcos(y)y'=0
\end{equation*}
\begin{equation*}
M_y=(x^2e^x+2xe^x)cosy
\end{equation*}
\begin{equation*}
N_x=cosy(2xe^x+x^2e^x)
\end{equation*}
\begin{equation*}
M_y=N_x\Rightarrow exact
\end{equation*}
\begin{equation*}
\varphi_y=N=x^2e^xcosy
\end{equation*}
\begin{equation*}
\varphi=x^2e^xsiny+h(y)
\end{equation*}
\begin{equation*}
\varphi_x=M \Rightarrow (x^2e^x+2xe^x)siny=2xe^xsiny+x^2e^xsiny+h'(y)
\end{equation*}
\begin{equation*}
x^2e^xsiny+2xe^xsiny=2xe^xsiny+x^2e^xsiny+h'(y)
\end{equation*}
\begin{equation*}
h'(y)=0
\end{equation*}
\begin{equation*}
h(0)=0
\end{equation*}
\begin{equation*}
\varphi=x^2e^xsiny+0=x^2e^xsiny
\end{equation*}
\begin{equation*}
x^2e^xsiny=C
\end{equation*}