# Toronto Math Forum

## MAT244--2019F => MAT244--Test & Quizzes => Quiz-2 => Topic started by: Yingyingz on October 04, 2019, 02:00:04 PM

Title: TUT5103 Quiz2
Post by: Yingyingz on October 04, 2019, 02:00:04 PM
7.Find an integrating factor and solve the given equation
$$\underbrace{1}_{M}+(\underbrace{\frac{x}{y}-\sin(y)}_{N})y^{\prime}=0.$$

$M_y=0$$N_x=\frac{1}{y} \because M_y\neq N_x \therefore not exact R_1=\underbrace{M_y-N_x}_{M'}=\underbrace{0-\frac{1}{y}}_{N'}=-\frac{1}{y}$$\left. \begin{array} { l } { \mu = e ^ { - \int R_1 d r } = e ^ { - \int \frac { 1 } { y } } = e ^ { \operatorname { ln } | y | } = y } \\ { \therefore y + ( x - y \operatorname { sin } ( y ) ) y ^ { \prime } = 0 \quad \because M'_y = N'_x = 1 \quad \therefore \text{exact function}} \\{\therefore \exists\quad\varphi(x,y)\qquad s.t \,\varphi_x=M'=y}\\{ \psi = \varphi y d x = y x + h ( y ) }\\{ \varphi y = x + h ^ { \prime } ( y ) = x - y \operatorname { sin } ( y ) }\\{ h ^ { \prime } ( y ) = - y \operatorname { sin } ( y ) }\\{ h ( y ) = - \int y \operatorname { sin } ( y ) }\\{ = y \operatorname { cos } ( y ) - \operatorname { sin } ( y ) }\end{array} \right. \varphi=yx+y\cos(y)-\sin(y)=c$\$