Question

Find the general solution of the following

2x²y^''-xy^'+(x-5)y=0

Answer 1

Find the general solution of the following

2x²y^''-xy^'+(x-5)y=0

Solution:

The general solution of the equation

$x^2y''+axy'+(bx^n+c)y=0$, $n\neq 0$

is:

$y=x^\frac{1-a}{2}[C_1J_\nu (\frac2n\sqrt{b}x^\frac n2)+C_2Y_\nu (\frac2n\sqrt{b}x^\frac n2)]$ ,

where $\nu=\frac 1n\sqrt{(1-a)^2-4c}$ , $C_1$ and $C_2$ are arbitrary constants;

$J_\nu (z)$ and $Y_\nu(z)$ are the Bessel functions of the first and second kind.

References :

Kamke, E., Differentialgleichungen: Losungsmethoden und Losungen, I, Gewohnliche Differentialgleichungen, B. G. Teubner, Leipzig, 1977.

Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC, Boca Raton, 2003.

In our case:

$2x^2y''-xy'+(x-5)y=0$

divided by 2:

$x^2y''-\frac12xy'+(\frac12x-\frac52)y=0$

$a=-\frac12$ , $b=\frac12$ , $c=-\frac52$ , $n=1$, $\nu=\sqrt{\frac94+10}=\frac72$ .

Solution:

$y=x^\frac{3}{4}[C_1J_\frac72 (\sqrt{\frac{x}{2}})+C_2Y_\frac72 (\sqrt{\frac{x}{2}})]$ ,

where $C_1$ and $C_2$ are arbitrary constants;

$J_\nu (z)$ and $Y_\nu(z)$ are the Bessel functions of the first and second kind.

Answer: $y=x^\frac{3}{4}[C_1J_\frac72 (\sqrt{\frac{x}{2}})+C_2Y_\frac72 (\sqrt{\frac{x}{2}})]$ .