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[SOLVED] 2.2.4 de with xy'

karush

Well-known member
Jan 31, 2012
2,928
$\textsf{find the general solution}$
$$xy'+2y=e^x$$
$\textsf{divide thru by x}$
$$y' +\frac{2}{x}y=\frac{e^x}{x}$$
$\textsf{Find u(x)}$
$$\displaystyle u(x)=\exp\int\frac{2}{x} \, dx=e^{\ln x^2}=x^2$$
$\textsf{so far anyway..}$
 

karush

Well-known member
Jan 31, 2012
2,928
$\textsf{find the general solution}$
$$xy'+2y=e^x$$
$\textsf{divide thru by x}$
$$y' +\frac{2}{x}y=\frac{e^x}{x}$$
$\textsf{Find u(x)}$
$$\displaystyle u(x)=\exp\int\frac{2}{x} \, dx=e^{\ln x^2}=x^2$$
$\textsf{multiply thru with $x^2$}$
$x^2 y' +2xy=xe^x$
$(x^2 y)'=xe^x$
$\textsf{integrate}$
$\displaystyle x^2 y=\int xe^x \, dx =e^x(x-1)+c$
$\textsf{divide thru by $x^2$}$
$\displaystyle y=\frac{e^x}{x}-\frac{e^x}{x^2}+\frac{c}{x^2}$
hopefully