$(3t+1)y'' - (9t+6)y' + 9y = -3.$

Question

I am given this equation:

$$(3t+1)y'' - (9t+6)y' + 9y = -3.$$

Is this a non-linear equation?

How do I find a solution to the associated homogeneous equation in the form $y(t) = e^{\lambda t}$?

Answer 1

Little hint

Try $y=bt+c$ then $y'=b$ and $y''=0$

$$ (3t+1)y'' - (9t+6)y' + 9y = -3$$ $$ - (9t+6)b + 9(bt+c) = -3$$ We have $-9bt+9bt=0$ $$ -6b + 9c = -3$$ $$ 2b -3c = 1$$ $$ b = \frac {1+3c} 2$$ $$y= \frac {(1+3c)} 2 t+c=c(\frac 3 2 t+1)+ \frac t 2$$ You can try to find the other solution knowing this...