MAT244-2014F > Quiz 1

Q1 problem 1 (L5101)

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Victor Ivrii:

2.6 p. 102, # 25
Solve
\begin{equation*}
(3x^2y+2xy+y^3)+ (x^2+y^2)y'=0.
\end{equation*}

Roro Sihui Yap:

(3 x^2 y + 2xy + y^3) + (x^2 + y^2) yâ€²= 0\label{A}

Let $M(x,y) = 3(x^2)y + 2xy + y^3$, $N(x,y) = x^2 + y^2$. Then $M_y(x,y) = 3x^2 + 2x + 3y^2$,  $N_x(x,y) = 2x$.

Equation (\ref{A}) is not exact. Lets try to find an integrating factor $\mu=\mu(x)$ to make it exact.
$d\mu/dx = (M_y - N_x)\mu / N\implies d\mu/dx = (3x^2 + 2x + 3y^2 - 2x)\mu / N \implies d\mu/dx = 3(x^2 + y^2) \mu / (x^2 + y^2)\implies d\mu/dx = 3Î¼\implies d\mu/ \mu = 3 dx\implies \ln \mu = 3x \implies \mu = e^{3x}$

Now multiply  the equation(\ref{A}) by  $\mu = e^{3x}$

\bigl((3 x^2y + 2xy +y^3)e^{3x}\bigr)+ \bigl((x^2+y^2)e^{3x}\bigr)yâ€²= 0
\label{B}

Now  $M(x,y) = 3(x^2y + 2xy +y^3)e^{3x}$,  $N(x,y) =(x^2+y^2)e^{3x}$. Then $M_y(x,y) = (3 x^2 + 2x + 3y^2)e^{3x}$, $N_x(x,y) = (2x3+x^2) + 3y^2)e^{3x}$,
$M_y(x,y) = N_x(x,y)$. Therefore the equation is exact. No need to check: it is exact due to construction of $\mu$. V.I.

There is a  $\Psi(x, y)$ such that:
\begin{gather}
\Psi _x(x, y) = M(x,y) =3(x^2y + 2xy +y^3)e^{3x},\label{C}\\
\Psi _y (x, y) = N(x,y) = (x^2+y^2)e^{3x}.\label{D}
\end{gather}

Integrating (\ref{C})  we have $\Psi (x, y) = (x^2+ \frac{1}{3} y^3)e^{3x} + f(y)$. Using this, differentiate to get
$\Psi_y (x, y) =(x^2+y^2)e^(3x) + f'(y)$. Easier to start from (\ref{D}) V.I.

Compare this with (\ref{D}): $f'(y) = 0$. $Meaning f(y) = C$, where $C$ is some constant
So,

\Psi (x, y) = (x^2+ \frac{1}{3}  y^3)e^{3x} +C=0
\$  is a solution.

Chang Peng (Eddie) Liu:
I'm having trouble typing out equations with proper format in this forum, so I did it in MSW and screenshot the work.. Apologies in advanced!

Victor Ivrii:
Roro. I rewrote your solution using superior math typesetting by MathJax (Javascript LaTeX/MathML parser). Everyone can quote my solution to see how it is done. Also fix your name.

Eddie. You got a karma as an exception: no need to post inferior technically (screenshot) solution after superior (typed) and correct. You could export your solution to LaTeX and after minimal corrections post "typed" solution. However code would be ugly and difficult to edit.

For everyone in the future: red is mine!

Chang Peng (Eddie) Liu:
Hi Prof. Ivrii,

It took me close to an hour to come up with that in MSW because I'm not used to typing out equations; so by the time I finished, Roro already posted it! But thank you!