I have doubts with a problem that goes like this:
Let's consider the second order linear differential equation
$ y''(x)+R(x)·y'+S(x)·y=Q(x) $
a) How must the change of variable $z = z(x)$ be so that the equation
is transformed into one of constant coefficients?b) Apply the above result to solve the equation:
$ y''(x)+(1+4e^{x})·y'+3e^{2x}·y=e^{2(x+e^{x})} $
I understand what the problem ask I don't know at all how to do it. I even wonder if the statement is right because the condition I get it's a bit abstract.
It's a Cauchy-Euler differential equation, so that:
$$ \frac {dy}{dx} = \frac {dy}{dz}· \frac {dz}{dx} = \dot yz'$$
$$ \frac {d^2 y}{dx^2} = … = \ddot{y}z'^{2}+ \dot yz'' $$
And the differential equation turns to:
$$ \ddot{y}z'^{2}+ \dot yz'' + R(x)·\dot y z' + S(x)y = Q(x) $$
$$ \ddot{y}z'^{2}+ \dot y (z'' + R(x)z') + S(x)y = Q(x) $$
If the equation should transformed into a constant coefficients equation, I think it means:
$$ z'' + R(x)z' = K $$
But I don'tknow what to do from here..
I supposse (in case this result is fine) I can't make K=0 (because it's just a special case) but this is the only value that let me develop a little but more this condition.
Please, anybody know whether this result is right or it has some errors?
How could I solve the problem please? (Not solve it completely, of course. Just a tip to let me continue doing it).
Thanks a lot.
Best Answer
$$\ddot{y}z'^{2}+ \dot y (z'' + R(x)z') + S(x)y =0$$ For this DE to be a constant DE you need: $$ \dfrac {S(x)}{z'^2}=c_1$$ And: $$\dfrac {z'' + R(x)z'}{z'^2}=c_2$$ Too many equations. So there must be some relation between $R(x)$ and $S(x)$.
The differential equation; $$y''(x)+R(x)·y'+S(x)·y=0$$ has no known solution. We can only solve it for some particular functions $R,S$.