In fact, with y+ z+ w= N, w= N- y- z so dw/dx= -dy/dx- dz/dx and d^2w/dx^2= -d^2y/dx^2- d^2z/dx^2 so d^2/dx^2+ d^2z/dx^2+ d^2w/dx^2= d^2y/dx^2+d^2z/dx^2- d^2y/dx^2- d^2z/dx^2= 0= y+ z+ w.
Any functions, y(x), z(x), w(x) that satisfy w(x)= -y(x)- z(x) will satisfy both equations!