d²y + P dy + Qy = R,
dx² dx

where P, Q, R are functions of x only. There is no method always effective; the main general result for such a linear equation is that if any particular function of x, say y1, can be discovered, for which

d²y1 + P dy1 + Qy1 = 0,
dx² dx

then the substitution y = y1η in the original equation, with R on the right side, reduces this to a linear equation of the first order with the dependent variable dη/dx. In fact, if y = y1η we have

dy= y1 dη+ η dy1and d²y= y1 d²η+ 2 dy1 dη+ η d²y1 ,
dx dxdx dx²dx² dxdx dx²

and thus

d²y + P dy+ Qy = y1 d²η+ (2 dy1+ Py1) dη+ ( d²y1+ P dy1+ Qy1)η;
dx² dxdx² dxdx dx²dx

if then

d²y1 + P dy1+ Qy1 = 0,
dx² dx

and z denote dη/dx, the original differential equation becomes