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How do you solve $y''+\frac{x}{2}y'+y=0$ where $y'=\frac{\partial y}{\partial x}$

What is confusing me is that once I say $g=y'$ I'm left with :

$g'+\frac{x}{2}g+y=0$

I can't use seperation by parts here. How would one proceed here? Thank you.

Best

Yukiko
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  • Use power series technique. – Mhenni Benghorbal Oct 28 '13 at 20:12
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    There are two unknown functions in your DE: $g'+\frac{x}{2}g+y=0$. To solve this for $g$ you have to know what $y$ is first. When you put $y$ in terms of $g$ you see that there is also an integral of $g$ there. – daulomb Oct 29 '13 at 11:55

1 Answers1

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In fact this question is similar to solution of ordinary differential equations:

Let $y=\int_Ce^{xu}K(u)~du$ ,

Then $(\int_Ce^{xu}K(u)~du)''+\dfrac{x}{2}(\int_Ce^{xu}K(u)~du)'+\int_Ce^{xu}K(u)~du=0$

$\int_Cu^2e^{xu}K(u)~du+\dfrac{x}{2}\int_Cue^{xu}K(u)~du+\int_Ce^{xu}K(u)~du=0$

$\int_C(u^2+1)e^{xu}K(u)~du+\int_C\dfrac{ue^{xu}K(u)}{2}d(xu)=0$

$\int_C(u^2+1)e^{xu}K(u)~du+\int_C\dfrac{uK(u)}{2}d(e^{xu})=0$

$\int_C(u^2+1)e^{xu}K(u)~du+\left[\dfrac{ue^{xu}K(u)}{2}\right]_C-\int_Ce^{xu}~d\left(\dfrac{uK(u)}{2}\right)=0$

$\int_C(u^2+1)e^{xu}K(u)~du+\left[\dfrac{ue^{xu}K(u)}{2}\right]_C-\int_Ce^{xu}\dfrac{uK'(u)+K(u)}{2}du=0$

$\int_C(u^2+1)e^{xu}K(u)~du+\left[\dfrac{ue^{xu}K(u)}{2}\right]_C-\int_Ce^{xu}\dfrac{uK'(u)-(2u^2+1)K(u)}{2}du=0$

$\therefore uK'(u)-(2u^2+1)K(u)=0$

$uK'(u)=(2u^2+1)K(u)$

$\dfrac{K'(u)}{K(u)}=2u+\dfrac{1}{u}$

$\int\dfrac{K'(u)}{K(u)}du=\int\left(2u+\dfrac{1}{u}\right)du$

$\ln K(u)=u^2+\ln u+c_1$

$K(u)=cue^{u^2}$

$\therefore y=\int_Ccue^{u^2+xu}~du$

But since the above procedure in fact suitable for any complex number $u$ ,

$\therefore y_n=\int_{a_n}^{b_n}c_n(k_nt)e^{(k_nt)^2+xk_nt}~d(k_nt)={k_n}^2c_n\int_{a_n}^{b_n}te^{{k_n}^2t^2+k_nxt}~dt$

For some $x$-independent real number choices of $a_n$ and $b_n$ and $x$-independent complex number choices of $k_n$ such that:

$\lim\limits_{t\to a_n}t^2e^{{k_n}^2t^2+k_nxt}=\lim\limits_{t\to b_n}t^2e^{{k_n}^2t^2+k_nxt}$

$\int_{a_n}^{b_n}te^{{k_n}^2t^2+k_nxt}~dt$ converges

For $n=1$, the best choice is $a_1=0$ , $b_1=\infty$ , $k_1=\pm i$

$\therefore y=C_1\int_0^\infty te^{-t^2}\cos xt~dt$ or $C_1\int_0^\infty te^{-t^2}\sin xt~dt$

Hence $y=C_1\int_0^\infty te^{-t^2}\sin xt~dt+C_2\int_0^\infty te^{-t^2}\cos xt~dt$

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