Transformation of certain Differential Equattm^., $£)%:■** 



2 m 2 # y = 2 ;h 2 # D 2 « + 4 m 3 xT)u + 4>m 4 xu 



= (D 2 + 2 m D + 2 ro 2 ) 2 w 2 x u— 4 w 2 D u — 4 w 3 ft. 



Substituting these values in the given equation, and dividing 



the result by D 2 -f 2 m D + 2 nt 2 , a factor common to all the 



terms, we have 



&* x u + 2 mD x u + (2 p — *)Du + 2m 2 xu + {2p— 4)ww =0; 



d 2 u ,_ _ _ . (du \ 



or x -r-a + (2 v — 2 + 2 m x) \ -. \- mu) = 0, 



dx z v \« / 



which is similar to the given equation, p being changed into 

 p — 1. If therefore 



y=(D 2 + 2mD + 2 m 2 )* m, 

 we shall have for the determination of u the integrable equk- 



d*u „ du „ 



tion - r — +2m- 7 \-2m /i u = 0. 



ax* ax 



We will now apply the transformation y=z(D + m)u to a 

 few examples at once. This gives 



y5&* .rD 3 w + w.rD 2 ?< = D 3 xu-3D 2 u+mT> 2 xu-&mDu 

 ax* % 



= (D+m)(D 2 xu— 3D«) + «D« 

 =s(D + m)(D .r ?*— 2 m) + w w 



x-r- — xD^u + m xD u=D 2 xu—2 D u + m Dxu—mu 

 ax 



xy=xD m + ?»tm = D j?m— M + wa?M=(D + ?w)a: w — w. 

 Let 



* {8 +( - m+r ^ +»»•*}+? (s+^) =o - 



If the above values be substituted in this, the result will have 

 the factor D + JW common to all the terms, and will reduce to 



{: 



Consequently, if g = (D + my u, the proposed will reduce to 



i d?u , N du ~\ . ,. /du \ 



X \dT* +{m + r) dx + m rU J + {P " l) {dx + r V ^ 



d 2 u , , N </« 



Again, let* (g + fc|g +pg + »y=0. 



The same values substituted in this will give in like manner 



(d 2 u du\ . ,. du , 



* ( T-4 + »»-^)+(l»— 1) t- + ««=0. 

 W# 2 </#/ v dx 



Therefore y — (D + ot)p w will reduce it to 



