DIFFERENTIAL EQUATIONS 179 



8.312 General condition for an exact differential equation. Write: 



%>-y ^ = y". *3.y,., 



dx dx 2 ' dx n y 



In order that the differential equation: 



V(x,y,y',y",. . .' . . ,yM)-o, 

 be exact it is necessary and sufficient that 



W _ A (W\ , &_ (<*L\ ( ^ d- /dV\ 



dy dx (dy'J + dx* (dy") ~ 4 }H dx^ (fyto) = ' 



8.400 Linear differential equations of the second order. 

 General form: 



J2 + P ^ + Qy = R! 

 dx 2 dx 



where P, Q t R are, in general, functions of x. 



8.401 If a solution of the equation with R = o: 



y = w 

 can be found, the complete solution of the given differential equation is: 



y = ciw + dwfe-S* ~ + 



8.402 The general linear differential equation of the second order may be 

 reduced to the form: 



^ + Iv = 

 where: y = v 



T n i dp I 



1 = U -- --- Jr . 



2 dx 4 



8.403 The differential equation: 



d 2 y dy ~ 



d* +p d- x +Q >' = > 



by the change of independent variable to 



becomes: d 2 y 



-L 



By the change of independent variable. 



dz = QeS pd * dx, 



/n.2 Pdx __ L_ 



Qe u(zy 



it becomes; 



d ( i dy 



