Question

Consider the differential equation: \[x \frac{ dy }{ dx }-2y=0\]

Find a general solution to this differential equation that has the form y=Cx^n.

I wasn't sure how to do this. First I separated the terms (forget what thats called) so I had: \[\frac{ dy }{ 2y }=\frac{ dx }{ x }\] and integrated both sides giving me: \[\frac{ 1 }{ 2 }\ln(y)+c = \ln(x)+c\] but I am stuck as to what to do next. Am I doing something wrong so far?

Answer 1

By exponentiating both sides, I get y=2x+c, which doesn't seem to be correct.

Answer 2

integrating both sides
(1/2)ln(y)=ln(x)+ln(c)
ln(y)=(1/2)ln(x)+(1/2)ln(c)
y=e^((1/2)ln(x))+e^((1/2)ln(c))