Toronto Math Forum
MAT2442018S => MAT244Tests => Quiz2 => Topic started by: Victor Ivrii on February 02, 2018, 02:12:28 PM

Show that the given equation is not exact but becomes exact when multiplied by the given integrating factor. Then solve the equation.
$$
(x + 2) \sin(y) + x \cos(y)y' = 0,\qquad \mu (x, y) = xe^x.
$$

Let $$M(x, y) = (x + 2)\sin(y) \qquad \text{ and } \qquad N(x, y) = x\cos(y)$$
Then, $$\frac{\partial}{\partial y}M(x, y) = (x + 2)\cos(y) \qquad \text{ and } \qquad \frac{\partial}{\partial x}N(x, y) = \cos(y)$$
Note that $M_y \neq N_x$, so the equation is not exact. However, multiplying through by $\mu(x, y) = xe^x$, we get a new equation $$(x + 2)xe^x\sin(y) + x^2e^x\cos(y)y' = 0$$
We can see that this equation is exact, since $$\frac{\partial}{\partial y} (x + 2)xe^x\sin(y)= (x + 2)xe^x\cos(y) = \frac{\partial}{\partial x} x^2e^x\cos(y)$$
Therefore, there exists a function $\psi(x, y)$ such that \begin{align*}\psi_x(x, y) &= (x + 2)xe^x\sin(y) \tag{1} \\\psi_y(x, y) &= x^2e^x\cos(y) \tag{2}\end{align*}
Integrating (1) with respect to $x$, we get $$\psi(x, y) = \int (x + 2)xe^x\sin(y)\,\mathrm{d}x = x^2e^x\sin(y) + h(y)$$
for some function $h$ of $y$. Next, differentiating with respect to $y$, and equating with (2), we get $$\psi_y(x, y) = x^2e^x\cos(y) + h'(y) = x^2e^x\cos(y)$$
Therefore, $$h'(y) = 0 \implies h \text{ is constant }$$ Taking $h(y) = 0$, we get $$\psi(x, y) = x^2e^x\sin(y)$$
Thus, the solutions of the differential equation are given implicitly by $$x^2e^x\sin(y) = C$$