Toronto Math Forum
MAT244--2019F => MAT244--Test & Quizzes => Quiz-2 => Topic started by: Linqian Shen on October 04, 2019, 02:00:02 PM
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Determine whether the equation is exact, if it is exact find the solution
$$
(2xy^2+2y)+(2x^2y+2x)y^{\prime}=0
$$
$$
\begin{array} { l }
{M_y=4xy+2\quad N_x=4xy+2}\\
{M_y=N_x\rightarrow\text{exact}}\\
{G \varphi(x,y)\quad\text{such that}\quad \varphi(x)=M\quad\varphi(y)=N}
\end{array}
$$
$$\left. \begin{array} { l } { \varphi x = 2 x y ^ { 2 } + 2 y } \\ { \varphi = x ^ { 2 } y ^ { 2 } + 2 x y + h ( y ) } \\ { \varphi y = 2 x ^ { 2 } y + 2 x + h ^ { \prime } ( y ) = 2 x ^ { 2 } y + 2 x } \end{array} \right.$$
$$\left. \begin{array} { l } {h^{\prime}(y)=0}\\{h(y)=\text{constant}=c}\\ {x^{2} y ^ { 2 } + 2 x y = c } \end{array} \right.$$