Find $\;\lim_{x \to \infty}{\frac{f'(x)}{f(x)}}\;$ of the greatest value where f(x) is a solution to differential equation.
Choices :
$$ 1.\;\; y''+5y'-6=0, \;\;y(0)=-1,\;\;y'(0)=1$$
$$ 2. \;\;4y''-8y'+3y=0, \;\;y(0)=2,\;\;y'(0)=1$$
$$ 3.\;\; 4y''-4y'+y=0, \;\;y(0)=2,\;\;y'(0)=1$$
$$ 4.\;\; 2x^2y''+3xy'-y=0, \;\;x>0, \;\;y(1)=0,\;\;y(2)=1$$
I could find all solution to these DE and have the answer from them,
but it would be quite cumbersome.
Is there another way?
The first one is easy to integrate
Last one is also easy to integrate
$$2x^2y''+3xy'-y=0$$ $$2x^2y''+4xy'-xy'-y=0$$ $$(2x^2y')'-(xy)'=0$$ $$2x^2y'-xy=K_1$$ Which is a first order diff equation $$y'-\frac y {2x}=\frac {C_1}{x^2}$$ Use integrating factor $\mu (x)=x^{-1/2}$ $$(yx^{-1/2})'=\frac {C_1}{x^{5/2}}$$ After integrating, you finally get that : $$\boxed {y(x)=K_1\sqrt x+ \frac {K_2} x}$$