Here's the problem:(adsbygoogle = window.adsbygoogle || []).push({});

x^2y''-3xy'-12y=0

with initial conditions y(1)=0 and y'(1)=7

I'm supposed to solve for y in the form y=c1y1+c2y2

y1 = x^6 by inspection

Now to solve for y2

y2=y1v

v can be solved for by the equation

y1v''+(2y1'+py1)v'=0

where p is the function in front of y' when there is no function in front of y'':

x^2y''-3xy'-12y=0 divide by x^2

y''-3/xy'-12/x^2y=0

so p=-3/x

x^6v''+(2*6x^5+(-3/x)x^6)v'=0

x^6v''+(12x^5-3x^5)v'=0

x^6v''=-9x^5v'

so v'=e^(-x^2/18)

and v= -x/9(e^(-x^2/18)+c

so y2=x^6(-x/9(e^(-x^2/18)))

y2=-x^(7)/9*e^(-x^2/18)

Now y=c1y1+c2y2 and my initial conditions are y(1)=0 and y'(1)=7

c1*x^6+c2*-x^(7)/9*e^((-x^2)/18)

y'=c1y1'+c2y2'

= c1*6*x^5+

c2*((-7/9*x^6*e^((-x^2)/18)+((-x^7)/9*(-x/9)*e^((-x^2)/18)

Subbing in 1 for x for both equations I get

c1+c2*-1/9*e^(-1/18)

and

6c1+c2*(-7/9*e^(-1/18)+(-1/9*-1/9*e^(-1/18))

=

6c1+c2*(-63/81*e^(-1/18)-1/81*e^(-1/18)

=

6c1-64/81*e^(-1/18)c2

I solved for c1 in the first equation:

c1=c2*1/9*e^(-1/18)

and plugged that into the second equation

6*c2*1/9*e^(-1/18)-64/81*e^(-1/18)*c2=7

c2*e^(-1/18)*(54/81-64/81)=7

c2*e^(-1/18)*-10/81=7

c2=-81/10*7*e^(1/18)

so c1=(-81/10*7*e^(1/18))*1/9*e^(-1/18)

c1=-567/10*e^(1/18)*1/9*e^(-1/18)

=-567/90

Now I plug those back in the equation c1y1+c2y2=y and that's not the right answer. There's a good chance I messed up in there somehow, although I've double checked everything. Does anyone see what I did wrong, or better yet, an easier way to solve this problem. This way is just ridiculous.

Thanks a lot.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Second order diff eq with two variables

**Physics Forums | Science Articles, Homework Help, Discussion**