2
$\begingroup$

I am studying about the linear odes with non-constant coefficients.

I know the first order linear ode with non-constant coefficient

$$y^{'}(x)+f(x)y(x)=0 \tag{1}$$

has a general solution of the form

$$y=Ce^{-\int f(x) dx} \tag{2}$$

However, I am more interested in the case of linear second order odes with non-constant coefficients

$$y^{''}(x)+g(x)y^{'}(x)+f(x)y(x)=0 \tag{3}$$

I know that this equation does not have a closed form solution like $(2)$. However, I am interested in special cases of that.


Questions

$1$. Consider $(3)$, when $g(x)=0$, then we have

$$y^{''}(x)+f(x)y(x)=0 \tag{4}$$

Is Eq.$(4)$ a famous well-known equation? If YES, what is its name?

$2$. Does $(4)$ have a closed form solution like $(2)$?

$3$. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?

For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don't know any well-known equation with non-polynomial coefficients.

$\endgroup$
  • 1
    $\begingroup$ For question number 4, you might want to look into generalized wave equations. (Another link) $\endgroup$ – flawr Jul 3 '16 at 10:54
  • $\begingroup$ @flawr: Thanks, the wave equations link was helpful. But I think Another link is discussing a first order equation. :) $\endgroup$ – H. R. Jul 3 '16 at 11:02
  • 1
    $\begingroup$ eqworld.ipmnet.ru/en/solutions/ode.htm (HTH) $\endgroup$ – Giuseppe Negro Jul 5 '16 at 12:15
  • $\begingroup$ @GiuseppeNegro: Thanks that was helpful. :) $\endgroup$ – H. R. Jul 5 '16 at 12:19
2
$\begingroup$

Somewhere (but where ?) I already answer to a question quite the same as your question 1 (only change a sign in equation 4). By luck, I didn't remove my draft (copy below).

There is no general formula for the solutions of equation 4 in cases of any $f(x)$ since specific special functions are defined according to each specific case. All the more so for equation 3.

COPY :

$$y''(x)=f(x)y(x)$$

Case: $f(x)=c^2 \quad\to\quad y(x)=c_1e^{cx}+c_2e^{-cx}=c_3\cosh(cx)+c_4\sinh(cx)$

Case: $f(x)=-c^2 \quad\to\quad y(x)=c_1\cos(cx)+c_2\sin(cx)$

Case: $f(x)=x \quad\to\quad y(x)=c_1Ai(x)+c_2Bi(x) \quad $ Ary functions.

Case: $f(x)=x^2 \quad\to\quad y(x)=c_1 D_{-1/2}(\sqrt{2}x) +c_2 D_{-1/2}(i\sqrt{2}x) \quad $ Parabolic cylinder function.

Case: $f(x)=-\lambda^2 x^{\frac{1}{\nu}-2} \quad\to\quad y(x)=c_1 \sqrt{x}J_{\nu}(\lambda x) +c_2 \sqrt{x}Y_{\nu}(\lambda x) \quad $ Bessel functions.

Case: $f(x)=\lambda^2 x^{\frac{1}{\nu}-2} \quad\to\quad y(x)=c_1 \sqrt{x}I_{\nu}(\lambda x) +c_2 \sqrt{x}K_{\nu}(\lambda x) \quad $ Modified Bessel functions.

Case: $f(x)=-a+2b\cos(2x) \quad\to\quad y(x)= c_1C(a\:,\:b\:;\:x)+c_1S(a\:,\:b\:;\:x)$ Mathieu functions.

Case: $f(x)=\frac{A}{x^2}+\frac{B}{x}+C \quad\to\quad y(x)=e^{-\frac{\gamma}{2}x}x^{\frac{\beta}{2}}\left( c_1 M(\alpha\:,\:\beta\:;\:\gamma x)+c_2 U(\alpha\:,\:\beta\:;\:\gamma x) \right) \quad$ with $\begin{cases} \gamma=\pm2\sqrt{C}\\ \beta=1\pm 2\sqrt{A+\frac{1}{4}}\\ \alpha=\frac{\beta}{2}+\frac{B}{\gamma} \end{cases}\quad$ Kummer and Tricomi functions (confluent hypergeometric functions).

Etc.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.