# Solve the following $ydx +xdy=e^{-xy}\;dx$ if it cuts the Y- axis.

Toolbox:
• First order , first degree DE
• Variable separable : Variables of a DE are rearranged to separate then, ie
• $f_1(x)g_2(y)dx+f_2(x)g_1(y)dy=0$
• Can be written as $\large\frac{g_1 (y)}{g_2(y)}$$dy=-\large\frac{f_1(x)}{f_2(x)}$$dx$
• The solution is therefore $\int \large\frac{g_1(y)}{g_2(y)}$$dy=-\int \large\frac{f_1(x)}{f_2(x)}$$dx+c$
• Under the variable separable method, identify the following forms: $d(uv)=udv+vdu$
• and $d\bigg(\large\frac{u}{v}\bigg)=\large\frac{vdu-udv}{v^2}$
$ydx+xdy=e^{-xy}dx$ if it cuts the y-axis
Step:1
$d(xy)=e^{-xy}dx$
Step 2:
$e^{xy}d(xy)=dx$
$\int e^{xy} d(xy)=\int dx+c$
$e^{xy}=x+c$
Step 3:
If t intersects the $y-axis$, x=0
$\therefore c= e^0=1$
The solution is
$e^{xy}=x+1$