$(3x+2y)dx+(2x+y)dy=0$

• Menentukan apakah PD eksak

  \[M(x,y)=3x+2y \ \ \ \ N(x,y)=2x+y\] \[\begin{align*} \text{Eksak ketika } \frac{\partial M}{\partial y} &= \frac{\partial N}{\partial x}\\ \frac{\partial (3x+2y)}{\partial y}=2 & \ \ \ \frac{\partial (2x+y)}{\partial x}=2 \end{align*}\]

  Maka PD tersebut merupakan PD eksak

• Mencari solusi

  \[\begin{split} (3x+2y)dx+(2x+y)dy=0 \\ 3xdx+2ydx+2xdy+ydy=0 \\ \bbox[5px, border: 1px solid black]{\int 3x dx =\frac{3}{2}x^2 \\ }\\ \bbox[5px, border: 1px solid black]{\int(2ydx+2xdy)=2\int(ydx+xdy)=2xy}\\ \bbox[5px, border: 1px solid black]{\int y dy = \frac{1}{2}y^2}\\ \frac{3}{2}x^2+2xy+\frac{1}{2}y^2=C \end{split}\]