# [SOLVED] differential equation

• Sep 13th 2007, 06:18 PM
malaygoel
[SOLVED] differential equation
Solve
$(x^2 + y^2 + x)dx=-xydy$

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Malay
• Sep 13th 2007, 06:33 PM
Jhevon
Quote:

Originally Posted by malaygoel
Solve
$(x^2 + y^2 + x)dx=-xydy$

Keep Smiling
Malay

recall that exact differential equations are those of the form $M(x,y)~dx + N(x,y)~dy = 0$, where $\frac {\partial M}{\partial y} = M_y = N_x = \frac {\partial N}{\partial x}$

So this guy starts off looking like an exact equation, right, with $M(x,y) = x^2 + y^2 + x$ and $N(x,y) = xy$

but lo and behold, $M_y = 2y \ne N_x = y$. thus we need to find an integrating factor

Recall that we can find the integrating factor $\mu$ by the differential equation:

$\frac {d \mu}{dx} = \frac {M_y - N_x}{N} \mu$

so here we have:

$\frac {d \mu}{dx} = \frac {2y - y}{xy} \mu = \frac {\mu}{x}$

$\Rightarrow \mu (x) = x$

multiplying through by $x$ we obtain:

$\left( x^3 + xy^2 + x^2 \right)~dx + \left( x^2y \right)~dy = 0$

which is now an exact equation, since $M_y = 2xy = N_x$

can you continue?