1. ## differential eqns ( try one please)

-4x+y+(x+4y)y'=0

give value for k in which differential eqn is exact
10x^19*y^2+kx^20*y+(x+y)*x^20 dy/dx =0

dy/dx=2x^2-3x-2+(2/x)*y

dy/dx=(6-(x+3)y)/x+2

2. Originally Posted by samdmansam
-4x+y+(x+4y)y'=0

give value for k in which differential eqn is exact
10x^19*y^2+kx^20*y+(x+y)*x^20 dy/dx =0

dy/dx=2x^2-3x-2+(2/x)*y

dy/dx=(6-(x+3)y)/x+2

$\displaystyle \frac{dy}{dx}=2x^2-3x-2+\frac{2}{x}y$

$\displaystyle \frac{dy}{dx}-\frac{2}{x}y=2x^2-3x+2$

so we need our integration factor

$\displaystyle I(x)=e^{\int \frac{2}{x}dx}=e^{2 \ln(x)}=e^{\ln(x)^2}=x^2$

$\displaystyle x^2\frac{dy}{dx}-2xy=2x^4-3x^3-2x \iff \frac{d}{dx}\left[ x^2y \right]= 2x^4-3x^3-2x$

integratin we get

$\displaystyle x^2y=\frac{2}{5}x^5-\frac{3}{4}x^4-x^2+c$

$\displaystyle y=\frac{2}{5}x^3-\frac{3}{4}x^2-1+cx^{-2}$