Question

Solve the equation x^3 y^'" + 5x^2 y" + 7xy' + 8y = 0

Answer 1

Make the substitution
`t=\ln x`, then compute the derivatives of
`y` with respect to
`t` via the chain rule.

• First derivative

`(\mathrm dy)/(\mathrm dx)=(\mathrm dy)/(\mathrm dt)(\mathrm dt)/(\mathrm dx)`

`\implies(\mathrm dy)/(\mathrm dx)=\frac1x(\mathrm dy)/(\mathrm dt)`

• Second derivative

Let
`f(t)=(\mathrm dy)/(\mathrm dt)`.

`(\mathrm d^2y)/(\mathrm dx^2)=(\mathrm d)/(\mathrm dx)\left[\frac fx\right]=(x(\mathrm df)/(\mathrm dx)-f)/(x^2)`

`(\mathrm df)/(\mathrm dx)=(\mathrm df)/(\mathrm dt)(\mathrm dt)/(\mathrm dx)=\frac1x(\mathrm d^2y)/(\mathrm dt^2)`

`\implies(\mathrm d^2y)/(\mathrm dx^2)=\frac1{x^2}\left((\mathrm d^2y)/(\mathrm dt^2)-(\mathrm dy)/(\mathrm dt)\right)`

• Third derivative

Let
`g(t)=(\mathrm df)/(\mathrm dt)=(\mathrm d^2y)/(\mathrm dt^2)`.

`(\mathrm d^3y)/(\mathrm dx^3)=(\mathrm d)/(\mathrm dx)\left[(g-f)/(x^2)\right]=(x^2\left((\mathrm dg)/(\mathrm dx)-(\mathrm df)/(\mathrm dx)\right)-2x(g-f))/(x^4)`

`(\mathrm dg)/(\mathrm dx)=(\mathrm dg)/(\mathrm dt)(\mathrm dt)/(\mathrm dx)=\frac1x(\mathrm d^3y)/(\mathrm dt^3)`

`\implies(\mathrm d^3y)/(\mathrm dx^3)=(x^2\left(\frac1x(\mathrm dg)/(\mathrm dt)-\frac1x(\mathrm df)/(\mathrm dt)\right)-2x(g-f))/(x^4)=\frac1{x^3}\left((\mathrm d^3y)/(\mathrm dt^3)-3(\mathrm d^2y)/(\mathrm dt^2)+2(\mathrm dy)/(\mathrm dt)\right)`

Substituting
`y(t)` and its derivatives into the ODE gives a new one that is linear in
`t`:

`\left((\mathrm d^3y)/(\mathrm dt^3)-3(\mathrm d^2y)/(\mathrm dt^2)+2(\mathrm dy)/(\mathrm dt)\right)+5\left((\mathrm d^2y)/(\mathrm dt^2)-(\mathrm dy)/(\mathrm dt)\right)+7(\mathrm dy)/(\mathrm dt)+8y=0`

`(\mathrm d^3y)/(\mathrm dt^3)+2(\mathrm d^2y)/(\mathmr dt^2)+4(\mathrm dy)/(\mathrm dt)+8y=0`

`y'''+2y''+4y'+8y=0`

which has characteristic equation

`r^3+2r^2+4r+8=(r+2)(r^2+4)=0`

with roots
`r=-2` and
`r=\pm2i`, so that the characteristic solution is

`y_c(t)=C_1e^(-2t)+C_2\cos2t+C_3\sin2t`

Replace
`t=\ln x` to solve for
`y(x)`:

`y_c(x)=C_1e^(-2\ln x)+C_2\cos(2\ln x)+C_3\sin(2\ln x)`

`\boxed{y(x)=(C_1)/(x^2)+C_2\cos(2\ln x)+C_3\sin(2\ln x)}`