Differential equations - reduction of order
given , where y1 =

With reduction of order, we assume a solution of the form , with . Then




and substituting into the ODE gives





Let , so that . This gives the linear ODE



This equation is also separable, so you can write



Integrating both sides with respect to gives




Next, solve for by integrating both sides again with respect to .





And finally, solve for .



and note that is already taken into account as part of , so this is the general solution to the ODE.