In this problem we determine conditions on p and q that enable Eq. (i) to be transform ed into an equation with constant coefficients by a change of the independent variable. Let x=u(t)be the new independent variable, with the relation between x and t to be specified later.(a) Show that dydt=dxdtdydx, d2ydt2=dxdt2d2ydx2+d2xdt2dydx.(b) Show that the differential equation (i) becomes dxdt2d2ydx2+d2xdt2+p(t)dxdtdydx+q(t )y=0.(iv)(c) In order for Eq. (iv) to have constant coefficients, the coefficients of d2y/dx2 and of y must be proportional. Ifq(t)>0, then we can choose the constant of proportionality to be 1; hencex=u(t)=[q(t)]1/2dt.(v)(d) With x chosen as in part (c), show that the coefficient ofdy/dxin Eq. (iv) is also a constant, provided that the expression q'(t)+2p(t)q(t)2[q(t)]3/2(vi)is a constant. Thus Eq. (i) can be transformed into an equation with constant coefficients by a change of the independent variable, provided that the function(q'+2pq)/q3/2 is a constant. How must this result be modified if q(t)<0?