# Toronto Math Forum

## MAT244--2019F => MAT244--Test & Quizzes => Quiz-2 => Topic started by: ZYR on October 06, 2019, 02:11:39 AM

Title: TUT0801
Post by: ZYR on October 06, 2019, 02:11:39 AM

$1 + (\frac{x}{y} - \sin(y))y' = 0$

Rewrite the equation, we have: $1dx + (\frac{x}{y} - \sin(y))dy = 0$

Let $M = 1$, and $N = \frac{x}{y}- \sin(y)$

Then $M_y = 0$, and $N_x= \frac{1}{y}$, where $M_y \neq N_x$
So, it is not exact.

Let $R_1 = \frac{M_y - N_x}{M} = \frac{-1}{y}$

So $\mu = e^{-\int R_1 dy} = e^{\int \frac{1}{y} dy} = e^{\ln{y}} = y$

Then multiply $\mu = y$ on both sides, we have

$y dx + (x - y\sin(y)) dy = 0$

and now it is exact since $M_y = 1 = N_x$

So $\exists\ \psi(x,y)$, such that $\psi_x = M = y$

$\psi = \int M dx = \int y dx = yx + h(y)$

Then, we can get $\psi_y = x + h'(y)$

Also, we know $\psi_y = N = x - y\sin(y)$, which implies that

$h'(y) = - y\sin(y)$

So $h(y) = \int h'(y) dy = \int - y\sin(y) dy$

By integrating by parts, we can know

$h(y) = y\cos{y} -\sin{y} + c$

Therefore, we can have

$\psi = yx + y\cos{y} -\sin{y} = c$