2
$\begingroup$

Please help with the following problem:

Let $ϕ$ be a solution of the equation $y''+a_1y'+a_2y=0$, where $a_1,a_2$ are constants. If $ψ(x) = e^{(\frac{a_1}{2})x}ϕ(x)$, show that $ψ$ satisfies an equation $y''+ky=0$, where $k$ is some constant. Compute $k$.

$\endgroup$
2
  • $\begingroup$ Do you mind sharing your own thoughts on the problem? $\endgroup$
    – mickep
    Feb 19, 2017 at 18:36
  • $\begingroup$ what have you done already? $\endgroup$ Feb 19, 2017 at 18:36

1 Answer 1

1
$\begingroup$

Hint: $$ψ(x) = e^{\frac{a_1}{2}x}ϕ(x)$$ $$ψ'(x) = \frac{a_1}{2}e^{\frac{a_1}{2}x}ϕ(x)+e^{\frac{a_1}{2}x}ϕ'(x)$$ and take second derivate $ψ''(x)$ then substitute in $\dfrac{y''}{y}$.

$\endgroup$
2
  • $\begingroup$ Substitute in to $y′′+ky=0$? $\endgroup$ Feb 19, 2017 at 18:44
  • $\begingroup$ You can substitute in $y''+ky$ and show it is zero for a $k$, but my offer is better, because you don't know what's $k$. $\endgroup$
    – Nosrati
    Feb 19, 2017 at 18:46

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .