MATH 308 Lecture 9

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Lecture Notes

Exact Equations

$\left({\frac {\sin {y}}{y}}-2\mathrm {e} ^{-x}\,\sin {x}\right)\,\mathrm {d} x+\left({\frac {\cos {y}+2\mathrm {e} ^{-x}\,\cos {x}}{y}}\right)\,\mathrm {d} y=0$

Recall that ${\frac {\partial M}{\partial y}}={\frac {\partial N}{\partial x}}$ in order for the differential equation to be considered exact

${\begin{aligned}{\frac {\partial M}{\partial y}}&={\frac {\cos {y}}{y}}-{\frac {\sin {y}}{y^{2}}}\\{\frac {\partial N}{\partial x}}&={\frac {2}{y}}\left(-\mathrm {e} ^{-x}\,\cos {x}-\sin {x}\,\mathrm {e} ^{-x}\right)\end{aligned}}$

Equation is not exact so we can stop here.

We multiply by the integrating factor $\mu (x,y)=y\,\mathrm {e} ^{x}$ to obtain

${\frac {\partial M}{\partial y}}={\frac {\partial N}{\partial x}}=\cos {y}\,\mathrm {e} ^{x}-2\sin {x}$

Now we differentiate $F(x,y)=\left(\sin {y}\,\mathrm {e} ^{x}-2y\,\sin {x}\right)\,\mathrm {d} x+\left(\cos {y}\,\mathrm {e} ^{x}+2\cos {x}\right)\,\mathrm {d} x=0$ with respect to $y$

${\begin{aligned}{\frac {\partial F}{\partial y}}&=\cos {y}\,\mathrm {e} ^{x}+2\cos {x}+k'(y)\\\ldots \end{aligned}}$

Implicit solution: $\sin {y}\,\mathrm {e} ^{x}+2y\,\cos {x}+C=0$

$y'(t)=f(y)=\mathrm {e} ^{y}-1$

Equilibrium solution:

${\begin{aligned}f(y)&=0\\\mathrm {e} ^{y}-1&=0\\\mathrm {e} ^{y}&=1\\y&=\ln {1}\\y&=0\end{aligned}}$

Is it stable/unstable/semi-stable?

for $y<0$ , the function is decreasing.
for $y>0$ , the function is increasing.
unstable.

Phase line:

 -----------<-----------|----------->-----------
          y < 0         0         y > 0
         y' < 0 (dec)            y' > 0 (inc)
        y'' < 0 (ccdn)          y'' > 0 (ccup)

Concavity?

${\begin{aligned}y'&=f(y)\\y''&=f'(y)\,f(y)\\f'(y)&=\mathrm {e} ^{y}>0\end{aligned}}$

Linear vs Nonlinear

Linear	non-linear
$y'+P(x)y=g(x)\quad y(x_{0})=y_{0}$	$y'=f(x,y)\quad y(x_{0})=y_{0}$
$P(x)$ and $g(x)$ are continuous on an open interval $I$ $x_{0}\in I$ , $y_{0}\in \mathbb {R}$	$f(x,y)$ and ${\tfrac {\partial f}{\partial y}}$ are continuous on $I\times J$ (product of 2 intervals) $x_{0}\in I$ , $y_{0}\in J$
solution to init. val. problem exists solution is unique solution exists on $I$	solution to init. val. problem exists solution is unique solution exists on interval included in $I$