MAT244--2019F > Quiz-2

TUT0601 Quiz2


Mingzhu Yu:
find the value of b for which the given equation is exact, and then solve it using that value of b.
$$\therefore M(x,y)=ye^{2x}+x$$
since the differential equation to be exact
we get b=1, then put b=1 in the differential equation.
$$M=ye^{2xy}+x, N=xe^{2xy}$$
Since the differential equation is exact then there is a function $\varphi(x,y)$ such that $\varphi_x(x,y)=M(x,y)$ and $\varphi_y(x,y)=N(x,y)$
$$\because\varphi_y=N\qquad\therefore xe^{2xy}+h'(y)=xe^{2xy}$$
$$\therefore h'(y)=0\qquad\therefore h(y)=c$$
General solution  $\varphi=\dfrac{e^{2xy}}{2}+\dfrac{x^2}{2}=c$


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