# Use the method of reduction of order to find the general of the differential equation

Use the method of reduction of order to find the general of the differential equation
$t\left(t+3\right)y{}^{″}-3\left(t+2\right){y}^{\prime }+3y=0$

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Sally Cresswell

Solve $t\frac{{d}^{2}y\left(t\right)}{{dt}^{2}}\left(t+3\right)-3\frac{dy\left(t\right)}{dt}\left(t+2\right)+3y\left(t\right)=0:$
Let
Simplify:
$t\frac{{d}^{2}v\left(t\right)}{{dt}^{2}}\left({t}^{2}+5t+6\right)+\frac{dv\left(t\right)}{dt}\left(-{t}^{2}-6t-12\right)=0$
Let $\frac{dv\left(t\right)}{dt}=u\left(t\right)$, which gives $\frac{{d}^{2}v\left(t\right)}{{dt}^{2}}=\frac{du\left(t\right)}{dt}:$
Solve for $\frac{du\left(t\right)}{dt}:$
$\frac{du\left(t\right)}{dt}=\frac{\left({t}^{2}+6t+12\right)u\left(t\right)}{t\left({t}^{2}+5t+6\right)}$