2022-01-20

solving second order linear ODE
$\frac{a}{x}{y}^{\prime }\left(x\right)+\frac{1}{2}y{}^{″}\left(x\right)=0$
where a is a constant. Also how can I solve this equation if the right hand side is different than zero?

Papilys3q

Expert

Step 1
Consider $z={y}^{\prime }$. Then this is just $\frac{a}{x}z\left(x\right)+\frac{1}{2}{z}^{\prime }\left(x\right)=0$. That is ${z}^{\prime }=-\frac{2a}{x}z$ which is separable.
Step 2
$\mathrm{ln}|z|=\int \frac{dz}{z}=\int -\frac{2a}{x}dx=-2a\mathrm{ln}|x|+{C}_{0}$ and so ${y}^{\prime }=z={C}_{1}{x}^{-2a}$.
Integrating yields $y=\frac{{C}_{1}}{1-2a}{x}^{1-2a}+{C}_{2}$

Karen Robbins

Expert

Step 1
Assuming a solution in an interval not containing $x=0$, lets rewrite the equation as:
$2a{y}^{\prime }+xy{}^{″}=0$
Now put ${y}^{\prime }=v,y{}^{″}={v}^{\prime }$ to get
$2av+x{v}^{\prime }=0$
Step 2
Separating variables produces
$\frac{2a}{x}=-\frac{{v}^{\prime }}{v}$
$2a\mathrm{log}Cx=-\mathrm{log}v$
$2a\mathrm{log}Cx=-\mathrm{log}v$
$K{x}^{-2a}={y}^{\prime }$

RizerMix

Expert