# Math Help - differential equations

1. ## differential equations

find a general solution to the differential equation

y*ln(x)-xy' =0

this is my work so far:
ylnx=x dy/dx
ylnx dx= xdy
(lnxdx)/x= dy/y
integral of (lnx)/x dx = integral of dy/y

???? = lny

i need help finding the integral of (lnx)/x dx
and then i need to solve for y

thank you!

2. Originally Posted by holly123
find a general solution to the differential equation

y*ln(x)-xy' =0

this is my work so far:
ylnx=x dy/dx
ylnx dx= xdy
(lnxdx)/x= dy/y
integral of (lnx)/x dx = integral of dy/y

???? = lny

i need help finding the integral of (lnx)/x dx
and then i need to solve for y

thank you!
Try $u = \ln x$ in $\int \frac{\ln x}{x}\, dx$

3. oh thank you! i was on winter break for 2 weeks and i guess i'm fuzzy on integrating

4. wait so what do i do after i get u=lnx and du= 1/x dx
is it 1/2 u^2 +c
1/2 (lnx)^2 +c ??

5. Originally Posted by holly123
wait so what do i do after i get u=lnx and du= 1/x dx
is it 1/2 u^2 +c
1/2 (lnx)^2 +c ??
Yep - that's right!

6. thanks!! so i have to inject e to get rid of ln and solve for y
would that be y= 1/2 x^2 + e^c

7. Originally Posted by holly123
thanks!! so i have to inject e to get rid of ln and solve for y
would that be y= 1/2 x^2 + e^c
You can't simplify like that. In general

$\left( \ln x \right)^n \ne \ln \left( x^n \right)$

and

$a \ln (x) \ne \ln (a x)$

8. for some reason this is really confusing me...i tried again and got y=x+ e^c

9. Originally Posted by holly123
for some reason this is really confusing me...i tried again and got y=x+ e^c
Since $\ln y = \frac{1}{2} \left( \ln x \right)^2 + c$

then the best you're going to get is

$y = e^{\frac{1}{2} \left( \ln x \right)^2} \cdot e^c = C e^{\frac{1}{2} \left( \ln x \right)^2}$ where $C = e^c$